1 DICTIONARY OF MATHEMATICAL TERMS DR. ASHOT DJRBASHIAN in cooperation with PROF. PETER STATHIS2 23 i INTRODUCTION This dictionary is specifically des...

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Meagan Veronica Barker

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i

INTRODUCTION

This dictionary is specifically designed for two-year college students. It contains all the important mathematical terms the students would encounter in any mathematics classes they take. From the most basic mathematical terms of Arithmetic to Algebra, Calculus, Linear Algebra and Differential Equations. In addition we also included most of the terms from Statistics, Business Calculus, Finite Mathematics and some other less commonly taught classes. In a few occasions we went beyond the standard material to satisfy curiosity of some students. Examples are the article ”Cantor set” and articles on solutions of cubic and quartic equations. The organization of the material is strictly alphabetical, not by the topic. There are approximately a total of 1200 entries in this dictionary. Some of them are just simple one-two sentence definitions while others are pretty detailed articles with definitions and even worked examples. All the entries are in lowercase letters unless they contain a proper name. In text, however, some terms might appear in capitalized form in expressions such as Fundamental Theorem of Calculus, Mean value theorem and so on. There are also about eighty illustrations of major functions and geometric figures. Most of the pictures are created by the author and others are imported from open internet sources such as Google and Wikipedia. Because this is an electronic book and not a traditional hard copy book it is worth writing a few words how we think it should be used the best. The last few years brought lots of

changes in how students work with the books and treat them. More and more students opt for electronic books and ”carry” them in their laptops, tablets, and even cellphones. This is a trend that seemingly cannot be reversed. This is why we have decided to make this an electronic and not a print book and post it on Mathematics Division site for anybody to download. Here is how we envision the use of this dictionary. As a student studies at home or in the class he or she may encounter a term which exact meaning is forgotten. Then instead of trying to find explanation in another book (which may or may not be saved) or going to internet sources they just open this dictionary for necessary clarification and explanation. Why is this a better option in most of the cases? First of all internet search usually results in multiple, sometimes hundreds of choices and that already creates problems. Second, many of these sources are rather general and do not have in mind specifically the level of the students in community colleges. Many of them are actually designed with specialists in mind and contain material that might confuse the student rather than help. As an example, if a student needs to recall the topic ”Partial fractions” and ends up in Wikipedia then the explanation would be a very long, general, and confusing one containing lots of unnecessary material. The total number of pages corresponding to that particular entry in Wikipedia (after converting into Word format) is about 12 pages. In contrast, the explanation in this dictionary is very concise, contains only the necessary cases and fits on four Word pages, including

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worked examples of all cases. Traditional dictionaries use italics to indicate that the given word has its own entry. Electronic books present additional opportunity for the same purpose and especially for quick finding of the indicated terms, namely hyperlinks. In this book we use a combination of both methods and it is worth to clarify further how and when each of these approaches used in our dictionary. As the reader goes through some article in the dictionary he or she might need explanation, clarification, or definition of certain terms used in that article. When we thought that it is important and beneficial to the reader to refresh the memory and clarify that term in order to understand the article at hand we created a hyperlink allowing immediately jump to that term without scrolling through multiple pages. This is similar to most internet sources, such as above mentioned Wikipedia. For example, if the reader sees the expression characteristic equation, then by simply clicking at that word he will jump to page 20 of the text where that term is defined and explained. It is important to note however that not all the terms are hyperlinked and there are multiple reasons for that. First of all, that would make many articles very difficult to read because of abundance of hyperlinks and would destruct reader from the actual article itself. In addition, there are many terms so common that it is obvious that there are specific articles for them. These terms include, but not limited to, words such as Number, Function, Polynomial, Equation, Integral, Derivative, Matrix, etc. In rare cases, however, even some of these terms are hyperlinked by the reason explained above. Be-

cause of these in many cases we just used the traditional approach of emphasizing the terms with italics and that also requires some explanation. If the word is in italics then that means one of the following : 1) There is an article for that particular term but in author’s view it is not crucial to jump there immediately in order to understand the current article; 2) There may or may not be an article for that term but the definition is contained in this current article the reader is on; 3) There is an article for that term but it is so close (might be the very next or just the previous one) there is no need of hyperlink to get there. We hope that this hybrid approach would make the use of dictionary as easy as possible.

Dr. Ashot Djrbashian Mathematics Division Glendale Community College Glendale, CA

1

A Abel’s formula Also called Abel’s theorem, for differential equations. Let y1 and y2 be solutions of second order equation L[y] = y 00 + p(t)y 0 + q(t)y = 0, absolute value For a real number r, the nonnegative number |r| given by the formula n r if r ≥ 0 |r| = −r if r < 0 Absolute value of a real number indicates the distance between the point on the number line where c is a constant that depends on solutions corresponding to that number and the origin. y1 , y2 only. absolute value function The function f (x) = |x|, defined for all real values x. abscissa In the plane Cartesian coordinate system, the name of the x-axis. where both p and q are continuous functions on some interval I. Then by Abel’s theorem the Wronskian of the equation is given by the formula Z W (y1 , y2 ) = c exp − p(t) dt ,

absolute maximum and minimum (1) For functions of one variable. A number M in the range of a given function f (x), such that f (x) ≤ M for all values of x in the domain of f is the absolute maximum. Similarly, a number m in the range of a given function f (x), such that f (x) ≥ m for all values of x in the domain of f is the absolute minimum. If the function f is given on a closed interval [a, b], then to find absolute maximum and minimum values, the following steps to be taken: (a) Find all critical points (points, where the derivative f 0 is either zero or does not exist) of the function on the open interval (a, b) (b) Find the values of f at critical points and endpoints a and b. (c) Compare all these values to determine the largest and smallest. They will be the absolute maximum and minimum values respectively. See also local maximum and minimum. (2) For functions of several variables definitions similar to the above are also valid. The critical points are also used to find these maximum and minimum points.

P∞ absolutely convergent series A series n=1 an is absolutely convergent if the formed with the Pseries ∞ absolute values of its terms n=1 |an | converges. If a series is absolutely convergent, then any rearrangement of that series is also convergent and the sum is the same as the original series. acceleration In physics, the rate of change of the velocity of a moving object. Mathematically, if the distance traveled by the object is given by the function s(t), then acceleration is the second derivative of this function: a(t) = s00 (t). acute angle An angle that measures between 0 and 90◦ in degree measure or between 0 and π/2 in radian measure.

2 dimensions: the same number of rows and columns. If A and B are two m × n-matrices, then their sum is defined to be the matrix C which has as elements the sums of elements of the given matrices in each position. Example: 1 1 −2 −2 2 3 −1 0 4 + 1 −3 −4 5 −2 0 −4 0 1 −1 3 1 = 0 −3 0 . addends In arithmetic, two or more numbers to be 1 −2 1 added. Similarly, in algebra, two or more expressions to be added. Subtraction of matrices is defined similarly. addition One of the four basic operations in arithmetic and algebra. By adding two (or more) numbers we determine the sum of those numbers. addition and subtraction of complex numbers If a + ib and c + id are two complex numbers, then their sum or difference is defined to be another complex number, where the real part is the sum or difference of the real parts of the given numbers and the imaginary part is the sum or difference of imaginary parts: (a + ib) ± (c + id) = (a ± c) + i(b ± d). addition and subtraction of fractions If ab and c d are two numeric fractions, then their sum or difference is formally defined to be the fraction ad ± cb . bd

addition and subtraction trigonometry, the formulas

formulas

In

cos(φ + θ) = cos φ cos θ − sin φ sin θ cos(φ − θ) = cos φ cos θ + sin φ sin θ sin(φ + θ) = cos φ sin θ + sin φ cos θ sin(φ − θ) = cos φ sin θ − sin φ cos θ tan φ + tan θ tan(φ + θ) = 1 − tan φ tan θ tan φ − tan θ 1 + tan φ tan θ Similar formulas also exist for other trigonometric functions but are very rarely used. tan(φ − θ) =

In practice, to make calculations simpler, we first find the least common denominator (LCD) of two denominators of given fractions, substitute both fractions by equivalent fractions with the found LCD, and then add or subtract these two fractions. Example: to add 83 + 56 , we find the LCD of 8 and 6, which is 24, 29 9 + 20 substitute the given fractions by 24 24 = 24 . addition and subtraction of functions Let f (x), g(x) be two functions. Then the ”sum” function (f + g)(x) is defined to be the sum of their values: (f + g)(x) = f (x) + g(x). Similarly, (f − g)(x) = f (x) − g(x). addition and subtraction of matrices The sum of two matrices is defined only if they have the same

addition and subtraction of vectors Let u = (u1 , u2 , · · · , un ) and v = (v1 , v2 , · · · , vn ) be two vectors in some n-dimensional vector space V . To add or subtract these two vectors, we just add or subtract their corresponding coordinates: u ± v = (u1 ± v1 , u2 ± v2 , · · · , un ± vn ).

3 In case of vectors in the plane, addition of vectors has very simple geometric interpretation. It could be described by the triangle law (also called parallelogram law ): Place the initial point of the second vector on the terminal point of the first vector. The vector connecting the initial point of the first one with the terminal point of the second will be the sum of these vectors.

which is the leg, not the hypotenuse of the triangle.

(2) In the more general case when two events might have a common outcome, the addition rule is

Airy’s equation The second order equation

adjoint matrix Let A = (aij ) be a square matrix with real or complex entries and denote by Cij the cofactor of the element aij . Then the matrix made up by all cofactors, C11 C12 . . . C1n C21 C22 . . . C2n . .. .. .. addition property of equality Let A, B, C be . . any algebraic expressions. If A = B, then A + C = Cn1 Cn2 . . . Cnn B + C. is called the adjoint of the given matrix and denoted addition property of inequalities Let A, B, C by adj(A). The adjoint matrix allows computation of be any algebraic expressions. If A ≤ B, then A+C ≤ the inverse of the matrix A by the formula B + C. 1 adj(A). A−1 = addition rule for probabilities (1) If two events, det(A) A and B, are disjoint, then the addition rule is For a more practical way of determining the inverse matrix see the corresponding entry. P (A or B) = P (A) + P (B).

P (A or B) = P (A) + P (B) − P (A and B).

y 00 − xy = 0,

−∞ < x < ∞.

See series solution for exact form of the solution of this equation.

Note also that the set {A or B} is the union of the sets A and B and is denoted by A ∪ B and the set algebra One of the main branches of mathematics, {A and B} is the intersection of that sets and is that deals with numbers, variables, equations, etc. and operations with them. In a more general setting, denoted by A ∩ B. the developments of classical algebra, namely linear additive identity An element e of some set S,that algebra and abstract algebra, deal with vectors, mahas the property a + e = a for any element a. In the trices and structures such as groups, rings, fields and number system (real or complex), e = 0. others. additive inverse For a given element a of some set algebra of matrices The term is understood as S, the element b with the property a + b = e, where the collection of rules for operations with matrices: e is the additive identity. In the number system (real addition, scalar multiplication, multiplication of maor complex), e = 0 and b = −a. trices, and inverses of matrices. adjacent angle Two angles that have common algebraic equation An equation involving just vertex, one common side but no common interior algebraic function. Most common example is an alpoints. gebraic equation involving a polynomial: P (x) = 0 adjacent side Any of two sides in a triangle which or P (x1 , ..., xn ) = 0, where P is a polynomial of one make up the angle, are called adjacent. The other or n variables respectively. Example: 4x5 − 3x4 + side of the same triangle is the opposite side. In the x2 − 3x + 5 = 0. trigonometry of right triangles, this term is reserved Additionally, algebraic equations include also ratiofor one side of acute angle only, namely the side nal and radical equations.

4 algebraic expression An expression which involves numeric constants, one or more variables and only the algebraic operations of addition, subtraction, multiplication, division, and root extraction.

b and c are alternating angles.

algebraic function A function that involves the algebraic operations of addition, subtraction, multiplication, division, and root extraction. The function √ 3 3x2 + 2x − 5 f (x) = 4 x − 2x3 + 1 is an algebraic function but g(x) = log(sin x) is not. algebraic number A complex number z is an algebraic number if it satisfies a non-trivial polynomial equation P (z) = 0 for which the coefficients are rational numbers. It is known that many irrational numbers are not algebraic, in particular numbers π and e. These kind of numbers are called transcendental algebraic operation One of the basic operations of algebra: addition, subtraction, multiplication, division, or root extraction.

alternating harmonic series P∞ ries n=1 (−1)n+1 /n.

The convergent se-

alternating series A numeric series where the successive terms have opposite signs. alternating series estimation theorem Let S = P (−1)n−1 bn be the sum of an alternating series satisfying conditions

0 ≤ bn+1 ≤ bn , lim bn = 0. n→0 algorithm A method or procedure of solving a given problem or whole class of problems by a Then the remainder Rn of the series satisfies the inlimited number of standard operations. equality |Rn | = |S − Sn | ≤ bn+1 . See also error estimate for alternating series. P almost linear system In differential equations. alternating series test If ∞ n=1 an is an alternatLet A be a square matrix and x be a vector-function. ing series with The system of equations (i)|an+1 | < |an | for all n and x0 = Ax + g(x) (ii) lim |an | = 0, n→∞ is called almost linear, if the vector-function g is small then the series is convergent. compared to x, i.e., if alternative hypothesis A hypothesis (statement) that is contrary to the original (null) hypothesis. Notations for this hypothesis are Ha or H1 . The form of the alternative hypothesis varies depending on the form of the null hypothesis. If the null alternate angles When two parallel lines are hypothesis has the form H0 : µ = µ0 , then for the crossed by the third one (called a transversal), alternative hypothesis we have three possible forms: eight angles are formed. Two pairs of non-adjacent Ha : µ 6= µ0 , µ < µ0 , or µ > µ0 . Accordingly, interior angles are called alternate. Alternate angles we have two-sided (or two-tailed), left-sided, or are always equal. In picture below pairs a and d and right-sided alternative hypothesis. Some authors ||g(x)|| → 0, ||x||

x → 0.

5 also accept different forms of null hypothesis, specifically H0 : µ ≤ µ0 or µ ≥ µ0 . In this situation the alternative hypothesis will change accordingly. See also null hypothesis, hypothesis testing.

triangle has unique solution. (3) Here a = 12, b = 31, ; A = 20.5◦ . As in the previous case we find two solutions for the angle B : B1 ≈ 64.8◦ , B2 ≈ 115.2◦ . Unlike the previous case, here both are valid solutions. Continuing altitude of a triangle The perpendicular segment again as in the previous case we get also two from any vertex of a triangle to the opposite side (or solutions for the remaining elements. This means its continuation). Depending on the type of triangle that there are two possible triangles with the given (acute, right, obtuse), the altitude may or may not data (information). For the first one we have be inside the triangle. B1 ≈ 64.8◦ , C1 ≈ 94.7◦ , c1 ≈ 34.15, and for the second one B2 ≈ 115.2◦ , C2 ≈ 44.3◦ , c2 ≈ 23.93. ambiguous case One of the cases of solving See also solving triangles for other possible cases. triangles when two sides of a triangle and one angle not formed by these sides are given (the case SSA). amplitude of a graph For a graph of some peIn this case three outcomes are possible: triangle has riodic function the half of the difference between no solution, triangle has one definite solution, and absolute maximum and absolute minimum values. triangle has two definite solutions. In the following This notion is mostly used for trigonometric funcexamples the angles are denoted by A, B, C and tions. Example: For the graph of the function the sides are denoted by a, b, c and we follow the f (x) = 3 sin x the highest point (maximum) is 3 and convention that side a is opposite to angle A, side b the lowest point (minimum) is -3. So, the amplitude is opposite to angle B and side c is opposite to angle is a = (3 − (−3))/2 = 3. C. amplitude of trigonometric function For the functions sin and cos, the half of the difference between absolute maximum and absolute minimum values. Hence, for the function a sin(bt + c) + d, the amplitude will be |a|.

(1) Assume we know that a = 15, b = 25, A = 85◦ . Using the Law of sines we get sin A ≈ 1.66 > 1, a which is impossible. This means that it is not possible to construct a triangle with the given values. (2) Let a = 22, b = 12, A = 42◦ . Here the use of the Law of sines gives sin B ≈ 0.365 and solving this equation we get two solutions: B ≈ 21.4◦ and B ≈ 158.6◦ . The second angle is not possible because otherwise the sum of angles would have been greater than 180◦ , hence the first one is valid only. From here, C ≈ 116.6◦ . The remaining element of the triangle, the side c is found by another use of the Law of sines: c = a sin C/ sin A ≈ 29.4 and the sin B = b

analytic function For real-valued functions of one variable: the function is called analytic at some point x = a, if it is infinitely differentiable on some neighborhood of that point and its Taylor-MacLaurin series ∞ X f (n) (a) (x − a)n n! n=0 converges in a neighborhood of that point to the function itself. Not all infinitely differentiable functions are analytic. The function f (x) =

e−1/x 0

2

if if

x 6= 0 x=0

is infinitely differentiable, but all of its TaylorMacLaurin coefficients at the origin are zero and hence the series does not represent the function f .

6 analytic geometry One of the branches of geometry. Analytic geometry combines the geometric and algebraic methods by introducing coordinate systems, and presenting geometric objects with the help of algebraic functions and their graphs.

angle bisector

See bisector of an angle.

angle Geometric object, which consists of a point, called vertex and two rays coming out of this point. In trigonometry and calculus, angles usually are placed in standard position, when the vertex is placed at the origin of the Cartesian coordinate system, one side, called initial, coincides with the positive half of the x-axis and the other side, called terminal, can have arbitrary direction. In this position angles can be both positive and negative. The angles, where the terminal side is moved counterclockwise, are considered positive and clockwise direction is considered negative. The two main measuring units for angles are the degree measure and the radian measure. Two angles with the same terminal side are called coterminal and they differ in size by an integer multiple of a full circle, i.e. by multiple of 360◦ or 2π, depending on measuring unit. Angles traditionally are classified by their measure: An angle between 0 and 90 degrees is called acute, a 90 degree angle is called right, between 90 and 180 degrees – obtuse and 180 degrees– straight. Additionally, angles measuring 90, 180, 270 or 360 degrees are called quadrantal.

angle of elevation If an object is viewed from below, then the angle between the horizontal line and the line drown between the viewer and the object, is the angle of elevation.

angle of depression If an object is viewed from above, then the angle between the horizontal line and the line drown between the viewer and the object, is the angle of depression.

ANOVA Stands for Analysis of Variance. It is a collection of statistical methods designed to compare means of two or more groups of values and generalizes the normal distribution tests and t-tests. angular speed The rate of change of the angle as an object moves along a circle. If the angle is measured in radian measure and is denoted by θ, the radius of the circle is r, then the angular speed is given by the formula ω = θr . annihilator A differential operator is an annihilator for some function, if application of that operator to the function results in zero function. Example: The operator D2 + 4 is an annihilator for the function sin 2x, because (D2 + 4) sin 2x =

d2 sin 2x + 4 sin 2x = 0. dx2

angle between curves is defined to be the angle between tangent lines to these curves at the point of annual percentage rate The rate banks charge for borrowed money, or pay for the money deposited. Exintersection. pressed in percentages or equivalent decimal form. angle between vectors For vectors in inner prod- For example, annual percentage rate of 9 percent is uct spaces. If u, v are two vectors in the inner prod- translated into the numeric value 0.09, and APR 5.7 uct vector space V , then the angle θ between them is percent becomes 0.057. determined from the relation annulus A region bounded by two concentric circles of different radiuses. Also called ring or washer. hu, vi , cos θ = ||u||||v|| antiderivative Another name for indefinite integral.The antiderivative of a function is a new function where hu, vi is the inner product in V and ||u|| dewhich has that given function as its derivative. notes the norm (magnitude, length) of the vector. antidifferentiation Same as finding the indefinite angle between vector and plane is defined to integral. be the complement of the angle between that vector and the normal vector to the plane. antilogarithm For a number y and a base b, the

7 number x such that logb x = y. Same as exponential For values not too far from a this approximation function. gives satisfactory results. Also called linear approximation. approximate (1)The procedure of substituting for the exact value of a number with a value close to the approximation by Taylor polynomials If a exact value; (2) The procedure of substituting for the function f (x) can be expanded into a Taylor series values of a function by values of other, usually simpler, functions. Most often the approximations are done by polynomials or rational functions because of the simple nature of these functions. approximate integration The procedure of calculating definite integrals, when exact integration is impossible or difficult. There are various methods of approximate integration. For specific methods see midpoint rule, Simpson’s rule, trapezoidal rule, and approximate integration by Riemann sums below. approximate integration by Riemann sums By the definition of definite integral, it is the limit of Riemann sums, as the length of interval partitions approaches zero. Hence, each Riemann sum represents an approximate value of the given integral. Formally, let the function f (x) be defined on some interval [a, b] that converges to that function, then the partial sums of that series become approximations to the function. and form the left Riemann sum Let n ∞ X X f (j) (a) f (xi−1 )∆x. Rn = f (x) = (x − a)j j! i=1 n=0 Then the number Rn is the left endpoint approxima- be that expansion around the point x = a. If f is Rb tion of the integral a f (x)dx. The right endpoint bounded by M > 0 in the interval |x − a| ≤ d and approximation is defined similarly. ∞ X f (n) (a) R (x) = (x − a)n n approximation Collective term for different methn! j=n+1 ods allowing to approximate numbers, functions, and solutions of different types of equations, when exact is the remainder of the series, then numbers or solutions are not available. For more deM |x − a|n+1 tails see linear approximation, quadratic approxima|Rn (x)| ≤ , |x − a| ≤ d. tion, Newton’s method, and approximation by Taylor (n + 1)! polynomials below. This inequality, called Taylor’s inequality, tells that approximation by differentials For a given dif- the first n terms of the series represent a good apferentiable function f (x) around some point x = proximation for the function f . a, the simplest approximation is by a line passing The graph in this article shows Taylor approximathrough that point and with slope equal to the deriva- tions of the function sinx by Taylor polynomials of tive of the function at that point: f 0 (a). The equa- degree 1,2,3,4 and 5. tion of that line is given by approximation problems Different problems that come down to substituting the exact value of L(x) = f (a) + f 0 (a)(x − a).

8 some quantity by a value different but close to the actual value. arc Usually, a connected portion of a circle. In calculus, a connected portion of any smooth curve. arc length formula (1) For the circle. If the radius is r and an arc is subtended over the angle of measure θ (in radian measure), then the length of Archimedian spiral The polar curve r = θeiθ . the arc is given by the formula s = rθ. The parametric equation could be written as (2) For a general curve on the plane given by the x(t) = t cos t, y(t) = t sin t. function y = f (x) on interval [a, b]. The arc length is given by the formula Z

b

L=

p

1 + [f 0 (x)]2 dx.

a

(3) For a curve in R3 given by parametric equations x = x(t), y = y(t), z = z(t), where a ≤ t ≤ b, the arc length is given by a similar formula Z

b

s

dx dt

2

dy dt

2

dz dt

2

arcsine function The inverse of the sin x function, denoted either sin−1 x or arcsin x. Represents the angle θ on the interval [−π/2, π/2] such that sin θ = x. Formula for the parametric plane curve is similar, The domain of this function is [−1, 1] and the range with the last term missing. is [−π/2, π/2]. We have the following differentiation formula: arc length function The expression d 1 sin−1 = √ . dx Z xp 1 − x2 s(x) = 1 + [f 0 (t)]2 dt, L=

a

+

+

dt.

a

which comes out in calculations of arc lengths, is the arc length function. arccosine function The inverse of the cos x function, denoted cos−1 x or arccos x. Represents the angle θ on the interval [0, π] such that cos θ = x. The domain of this function is [−1, 1] and the range is arctangent function The inverse of the tan x [0, π]. We have the following differentiation formula: function, denoted tan−1 x or arctan x. Represents the angle θ on the interval (−π/2, π/2) such that tan θ = x. The domain of this function is (−∞, ∞) 1 d cos−1 = − √ . and the range is (−π/2, π/2). The inverse functions dx 1 − x2

9 cot−1 x, sec−1 x, csc−1 x are defined in a similar manner. We have the following differentiation formula: d 1 . tan−1 = dx 1 + x2

width w and length `, A = w · `. In the particular case when the length and width are equal, we get a square (of side a, for example) and the area is A = a2 . For a triangle with base b and height h, A = 21 b · h For a parallelogram with base a and height h, A=b·h For a trapezoid with two parallel bases a and b and height h, A = a+b 2 · h. In general, the area of any polygon could be calculated by dividing it into a number of triangles or rectangles and adding the areas of these smaller pieces. See also the corresponding definitions for pictures and notations.

area One of the basic notions of geometry used in many branches of mathematics. Indicates the area in polar coordinates If a curve is given with ”amount of plane” included in a certain plane region. the polar equation r = f (θ), where a ≤ θ ≤ b, then area between curves Let two curves be given by the area under that curve is equal to the functions f (x) and g(x). Then the area between Z 1 b that curves (on some interval [a, b]) is given by A= [f (θ)]2 dθ. 2 a Z b |f (x) − g(x)|dx. area of a sector The area of a sector of a circle a with radius r and the opening angle (in radian mea1 2 area function For a given non-negative function sure) θ is given by the formula A = 2 r θ. f (x) defined and continuous on some interval [a, b], area of a surface See surface area. the integral Z x area of a surface of revolution Assume we have A(x) = f (t) dt a surface obtained by revolving the smooth curve a y = f (x), a ≤ x ≤ b, about some line. Then, deis sometimes called area function, because it reprenoting by r the distance between arbitrary point x sents the area under the curve y = f (x) from point on interval [a, b] and the line, the surface area can be a to x. expressed as area of a circle For a circle with radius r, the Z b p area is given by the formula A = πr2 . S= 2πr 1 + (f 0 (x))2 dx. a

area of an ellipse The area of an ellipse is given by the equation area under the curve If a curve is given by the function f (x) ≥ 0, then the area under that curve 2 2 y x from point a to point b is given by the definite integral + = 1 a2 b2 Z b is equal to πab. Because the area does not change f (x)dx. during translation, the same value for the area is a also true for translated ellipse. area of geometric figures

For a rectangle with area under a parametric curve Suppose the

10 curve y = F (x), F (x) ≥ 0, is written in paramet- the inequality is as follows: Let a1 , a2 , · · · , an be any ric form x = f (t), y = g(t), α ≤ t ≤ β. Then the non-negative numbers. Then area under that curve (under the condition that this √ a1 + a2 + · · · + an n curve traverses only once as t increases from α to β) a1 · a2 · · · an ≤ n is given by the formula Z

β

A= α

g(t)f 0 (t)dt.

arithmetic growth A quantity that grows according to the arithmetic progression (see below) rule: an = a0 + (n − 1)d with some real number d.

argument of a complex number In the trigono- arithmetic sequence (progression) A sequence metric representation of a complex number z = of real numbers where the difference between two consecutive terms is the same. The general term of such reiθ = r(cos θ + i sin θ) the angle θ. sequence is given by the formula an = a1 + (n − 1)d, argument of a function In a function y = f (x) where d is the difference between two consecutive the independent variable x. Also, in a function of terms. The sum of the first n terms of arithmetic any number of variables, y = f (x1 , x2 , · · · , xn ), the sequence is given by the formula independent variables x1 , x2 , · · · , xn . n X a1 + an arithmetic The earliest and most fundamental n. ai = Sn = 2 branch of mathematics. Arithmetic deals with numi=1 bers and arithmetic operations on them: addition, subtraction, multiplication and division. In a more modern and wider sense, the term arithmetic is also associated quadratic form For any quadratic used as a synonym to number theory, which is the form, the associated quadratic form is the quadratic only part of arithmetic still subject to research activ- form with the same second degree terms and no first degree terms or constant. For quadratic form in two ity. variables ax2 +2bxy +cy 2 +dx+ey +f , the associated arithmetic average or mean For a positive inte- quadratic form would be ax2 + 2bxy + cy 2 . The case ger n, and given real numbers a1 , a2 , · · · , an , the mean of more variables is defined similarly. is (a1 + · · · + an )/n. Example: The average (mean) of the set of numbers {1, 3, −2, 5, 8, −4, 6, −5, 12} is association Any kind of relationship between two sets of variables or values. For example, in the func1 + 3 − 2 + 5 + 8 − 4 + 6 − 5 + 12 tion y = 2x + 1 the variables x and y are related by = 3. m= 8 the given formula and the values of y are associated with the values of x so that to each x the double of arithmetic expression An expression which in- that value plus one more is associated. Similarly, if volves numeric constants and the arithmetic opera- two sets of values forming ordered pairs are put into tions of addition, subtraction, multiplication, divi- a scatterplot, then by looking at the general direction we might see a positive association (bigger y’s sion, and natural exponents. Example: correspond to bigger x’s) or a negative one (bigger y’s correspond to smaller x’s). 72 − [(9 − 7)2 + 5 · 2] . 62 − 22 associative property (1) For addition: for any numbers a, b, c,complex or real, a+(b+c) = (a+b)+c. (2) For multiplication: for the same three numbers, arithmetic-geometric mean inequality The in√ a · (b · c) = (a · b) · c. a+b equality a · b ≤ 2 , which is true for any nonnegative numbers a and b. In the most general case astroid Let a circle of radius r roll inside another

11 circle of radius 4r. A point on a smaller circle will Augmented matrices are used to solve systems of linleave a trace that is called astroid. Parametric equa- ear equations with Gaussian or Gauss-Jordan elimitions are given by nation methods. x = cos3 θ,

y = sin3 θ.

Astroid is a special case of the hypocycloid. asymptote As a rule, a straight line with the property that the graph of a given function approaches it. There are three types of line asymptotes. Vertical asymptotes have the form x = a, a is real, horizontal asymptotes are of the form y = b, b is real and slant asymptotes are of the form y = ax + b with a 6= 0, ∞. Vertical asymptotes are characterized by the fact that limx→a± f (x) = ±∞ and the function f can never cross that line. For horizontal asymptotes we have limx→±∞ f (x) = b. Here the function may or may not cross the asymptote line. For the slant asymptotes we have limx→∞ [f (x) − (ax + b)] = 0 or the similar relationship with x → −∞. Here also, the function may or may not cross the line at some (possibly infinitely many) points. Slant asymptotes are also called oblique. In addition to line asymptotes also non-linear asymptotes are sometimes considered, such as quadratic or cubic.

autonomous equation First order differential equation of the form d x(t) = f (x(t)). dt These kind of equations describe particles which rate of motion depends only on their position but not of time. autonomous system equations.

System of autonomous

auxiliary equation For differential equations, the corresponding algebraic equation. For example, the equation ay 00 + by 0 + cy = 0 will have the auxiliary equation ar2 +br +c = 0. Also is called characteristic equation . average cost function Let x denote the number of units of any product by some company. Then C(x) denotes the cost function. The average cost function is defined to be the quantity c(x) = C(x)/x. This function is very important in economics.

average rate of change If a function f (x) is defined on some interval [a, b], then the average rate of asymptotic curve is a curve always tangent to an change of that function on any sub-interval [c, d] of asymptotic direction of the surface. An asymptotic the given interval is the quantity (f (c)−f (d))/(d−c). direction is one in which the normal curvature is zero. This quantity is used to find the instantaneous rate of change by making the interval [c, d] smaller and augmented matrix For the system of m equations eventually approaching its length to zero. with n unknowns average value of a function over an interval [a, b] a11 x1 + a12 x2 + · · · + a1n xn = b1 is defined by the formula Z b a21 x1 + a22 x2 + · · · + a2n xn = b2 1 f (x)dx. favg = ··························· b−a a am1 x1 + am2 x2 + · · · + amn xn = bm average velocity For a moving object the average asymptote of a hyperbola

See hyperbola.

the m × (n + 1)-matrix of all coefficients with the addition of constant terms column: a11 a12 . . . a1n b1 a21 a22 . . . a2n b2 . . .. .. .. .. . . . am1

am2

...

amn

bm

velocity in the time interval between t1 and t2 is determined by the formula v = (s2 − s1 )/(t2 − t1 ), where s1 and s2 are the distances traveled by the object at the moments t1 and t2 correspondingly. axiom A statement that is assumed to be true without proof, and used as a basis for proving other

12 statements, less obvious. Also is called postulate. axis (1) In Cartesian coordinate system, the name of lines that are used to calculate the position of a point in two, three or higher dimensional spaces. See also x-axis, y-axis, z-axis. The x-axis is also called abscissa and real axis and the y-axis is called ordinate and imaginary axis. (2) If a curve or a surface is symmetric with respect to some line, then this line is called axis of symmetry or rotation. Example: The y-axis is the axis of symmetry for the parabola y = x2 . (3) In polar coordinate system the ray coming out from the pole is called polar axis.

B back-substitution The last stage of the solution of systems of linear equations by Gaussian elimination method. After applying that method to a n × n system, we arrive to triangular system x1 + a12 x2 + a13 x3 + · · · + a1n xn = b1 x2 + a23 x3 + · · · + a2n xn = b2 ··························· xn−1 + an−1n xn = bn−1 x n = bn . Now, the back substitution consists of the following steps. First, we have xn = bn . Substituting back into the previous equation, we get xn−1 = bn−1 −an−1n bn . Continuing this way up, we eventually find all the values of variables x1 , x2 , · · · , xn−1 , xn , thus completing solution of the system. backward phase The second stage (or phase) of the solution of linear systems by Gaussian elimination. See back substitution above for details. bar graphs

Also called bar chart. Bar graphs are

one of many ways of visual representation of qualitative (categorical) data. In order to distinguish them from histograms, a little space is left between bars in the graph. The picture above shows the production numbers of some hypothetical company categorized

13 by months. base of exponential function In exponential function y = ax , where 0 < a < ∞, a 6= 1, and −∞ < x < ∞, the constant a is called the base and the variable x is called the exponent base of logarithm In logarithmic function y = logb x, where 0 < b < ∞, b 6= 1, and 0 < x < ∞, the constant b is called the base of logarithm.

N 52◦ E means the angle is 52 degrees measured from positive y-axis clockwise (to the East) and notation S47◦ W means that the angle is 47 degrees measured from negative y-axis clockwise (to the West). This method is used traditionally in marine navigation. The second method is mostly used in areal navigation and has only one starting direction (initial side of the angle), namely, the positive y-axis and the angles measured vary between 0 and 180 degrees.

basis Let S = {v1 , v2 , · · · , vn } be a subset of bell-shaped (Gaussian) curve The curve dethe linear vector space V . S is called a basis for scribing the normal distribution. Mathematically the V , if for any given vector u in V , there exists a function having that graph is given by the formula unique collection of scalars c1 , c2 , · · · , cn , such that (x − µ)2 1 u = c1 v1 + c2 v2 + · · · + cn vn . These numbers are , f (x) = √ exp − 2σ 2 σ 2π called coordinates (or components) relative to the given basis. A vector space has infinitely many bases and changing the basis changes also the coordinates of any given vector. Any basis is necessarily linearly independent. The number of the vectors in the set S is called dimension of the space V . Similar definition works also in cases when the dimension is not finite. A basis is called orthogonal, if the inner products of all possible pairs (of different vectors) in that basis are zeros: xi · xj = 0,

1 ≤ i, j ≤ n,

i 6= j.

If, additionally, xi · xi = 1, 1 ≤ i ≤ n, then the basis is called orthonormal. Example: The system of vectors e1 = (1, 0, 0), e2 = (0, 1, 0), e3 = (0, 0, 1) is a basis in the three dimensional Euclidean space R3 which is also orthonormal. The basis could be defined and found not only for the whole space, but also for any subspace of the given space V . For example, the nullspace of a matrix or transformation also has a basis because it is a subspace of the given space. Also, basis for a complex vector space is defined similarly with exception of a necessary change in the part of orthogonality. bearings Method of measuring angles used primarily in navigation. There are at least two approaches. First, the angles are measured with respect to vertical axis. In this situation all angles could be made acute if we measure starting from the ray that comes out from the origin and either goes up (North direction) or down (South direction). So, the notation

where µ and σ are the mean and standard deviation of the distribution respectively. The graph of this distribution looks like the one above. Bernoulli equations The first order non-linear equations y 0 + P (x)y = Q(x)y n . These equations can be transformed into linear equations w0 + P (x)w = Q(x) 1−n with the substitution w = y 1−n , n 6= 0, 1, and solved using integrating factors. Bessel equation In differential equations. The series of second order linear homogeneous equations x2 y 00 + xy 0 + (x2 − ν 2 )y = 0

14 with parameter ν. Depending on the value of this parameter we get Bessel equations of order zero, onehalf, one, two, and, in general, ν. The solutions are found by the method of series solution and heavily depend on the value of the parameter. See Bessel functions below. Bessel functions These functions arise during the series solution of the Bessel equations. Depending on the parameter ν, there could be many different Bessel functions. Additionally, this functions are categorized into first kind and second kind. The functions ∞ X (−1)n x2n J0 (x) = 1 + 22n (n!)2 n=1 and

∞ x X (−1)n x2n J1 (x) = 2 n=0 (n + 1)!n!22n

are called Bessel functions of the first kind of order zero and one respectively and they are the solutions of Bessel equations of order zero and one. The Bessel functions of second kind differ from functions of the first kind primarily by the presence of a logarithmic term. For example, the Bessel function of the second kind and order zero Y0 (x) is given by the representation " # ∞ X 2 (−1)m+1 Hm 2m x γ + ln J0 (x) + x π 2 22m (m!)2 m=1

and let u be some vector in V . Then the vector projW u is the best approximation from W to u in the sense that ||u − projW u|| ≤ ||u − w|| for any vector w that is in W . best fitting line A line that best describes the paired data collected from sample and presented by a scatterplot. See least squares regression line. bias In sampling or statistical experiment. If by some reason some statistic of collected data from sampling or experiment has the tendency of systematically differing from population parameter, then it is called biased. The reason of bias could be subjective (such as bad methodology of data collection) or objective (some statistics are always biased). biconditional statement A logical statement of the form ”If A then B and if B,then A”,or, equivalently, ”A if and only if B”. This statement also can be written symbolically as A ⇔ B. Examples can be: ”A triangle is equilateral if and only if it has three equal angles”, or ”A number is divisible by two if and only if it is even”. See also conditional statement.

binary numbers The binary numeral system represents numeric values using only two symbols, usually 0 and 1. In this system the sequence of whole where γ is the Euler’s constant and Hm is the mth numbers 0,1,2,3,... is represented by the following separtial sum of harmonic series. Now, the general so- quence: 0,1,10,11,100,101,110,111,1000,... The use of lution of the Bessel equation of order zero is given by this system is crucial in computers and other electronic devices because it allows to represent all real the function numbers (or their approximations) with the help of y = c1 J0 (x) + c2 Y0 (x) signal-no signal sequences. See also decimal numbers. binary operation An operation involving two mathematical objects. Examples of binary operabest approximation An approximation that protions are addition and multiplication of numbers, vides the closest possible value to a quantity we wish functions, matrices, etc. to approximate. Depending on nature of the object, best approximation will be found by a specific binomial A polynomial of one or more variables method. that has exactly two terms, called monomials. Examples are 2x + 1, 3x3 + 4x2 , 2x4 y 3 + 5xy 2 z. best approximation theorem Let W be a finitedimensional subspace of an inner product space V binomial coefficients In binomial expansion (see with arbitrary constants c1 , c2 .

15 below), the numeric coefficients n n! = , k k!(n − k)!

where α = α(α − 1)(α − 2) · · · (α − (n − 1))/n! n

where n ≥ 1 is an integer and 0 ≤ k ≤ n is another integer.

is the proper generalization of binomial coefficients (see also combinations) to the case of non-integer values of α. The series converges for all |x| < 1. binomial distribution Suppose we are conductIn the case when α is a positive integer, the series ing some experiment and the following conditions are reduces to finite binomial expansion. satisfies: bisector of an angle A ray, coming out of the 1) There are n repeated trials every time; 2) Each trial has exactly two possible outcomes, vertex of an angle, that divides the given angle into two equal parts. In a triangle, the bisector called success (S) and failure (F); 3) Probability of S is p and probability of F is of any angle is a segment coming out from the vertex, dividing the angle into equal parts. Similarly, q = 1 − p; 4) The random variable x is the number of successes bisectors could be defined for angles of any polygon. in n trials and takes values from 0 to n. In this case it is said that we have binomial setting. Binomial distribution is the distribution of probabilities in this setting and is given by the formula n k n−k p q , P (x = k) = k where nk = n!/k!(n − k)! is the binomial coefficient (see above). Binomial distributions serve as a good approximation for normal distributions. The mean and standard deviation of any binomial distribution could be calculated by very simple formulas: √ µ = np, σ = npq respectively. binomial expansion Also called Binomial Theorem. For any given real quantities x and y (fixed or variable) and any natural number n, the following expansion holds: n n 0 n n−1 1 n n−2 2 x y + x y + x y (x + y)n = 0 1 2 n n 0 n +··· + x1 y n−1 + x y n−1 n and nk , 0 ≤ k ≤ n are the binomial coefficients. binomial series ∞, the series

For a real number α, −∞ < α <

α

(1 + x) =

∞ X α n=0

n

xn ,

block of a matrix Let A be some n × m-matrix. We can divide this matrix into parts by separating certain rows and columns. Each part in this case is called a block. Example: The matrix 1 −1 −2 A = −1 3 4 5 −2 0 could be divided into four blocks as follows −1 B1 = (1) B2 = (−1 − 2) B3 = 5 3 4 B4 = . −2 0

16 This division is not unique and depends on what we equation need to do with this matrix. Operations done with p(x)y 00 − q(x)y + λr(x)y = 0 blocks of the matrix are called block manipulations. on the interval [0, 1] with be conditions Boolean algebra The system of rules of opera- α y(0) + α y 0 (0) = 0 and β y(1) + β y(1) = 0 1 2 1 2 tions with logical statements. This algebra involves is called Sturm-Loiville boundary value problem. Boolean variables, that take True and False values (or 1 and 0) and operations of conjunction (”and”), bounded function A function f , defined on some disjunction (”or”) and negation (”not”). Named af- interval I (finite or infinite), is bounded, if there exists a number M > 0, such that |f (x)| ≤ M for all ter George Boole. values x ∈ I. boundary For any set in the Euclidean space Rn , n ≥ 1, the boundary is the set of all boundary bounded sequence A sequence {an } of real or points (see below). Examples: (1) For the interval complex numbers is bounded if there exists a number [a, b] the boundary consists of two points a and b. (2) M > 0 such that |an | ≤ M for all values of n ≥ 1. For the ball x2 + y 2 + z 2 ≤ 1, the boundary is the bounded set A subset S of the real line R is sphere x2 + y 2 + z 2 = 1. bounded, if there is a positive number M > 0 such boundary conditions (1) For ordinary differential equations. If the equation is given on some finite interval [a, b], then the boundary consists of only two points a and b and the conditions are given on both points. The most general conditions could be written as α1 y(0) + α2 y 0 (0) = 0 and β1 y(1) + β2 y(1) = 0. (2) For partial differential equations. If the equation is given on some region D, then boundary is either some curve or a surface. Accordingly, the boundary conditions would be given on that boundary. boundary curve A curve that makes the boundary of some plane region. boundary point For a set D in the Euclidean space Rn , n ≥ 1, a point a ∈ D is a boundary point, if any ball with center at that point contains both points in D and outside of D. In case n = 1 the ball should be substituted by interval and in case n = 2 it should be a circle. Examples: (1) For the interval [a, b] each of the endpoints is a boundary point. (2) For the circle x2 + y 2 ≤ 1 the point (1, 0) is a boundary point.

that all elements x ∈ S satisfy |x| < M . If the set S is a subset of Rn , then similar definition applies with the |x| understood as the length of the point x ∈ Rn . boxplot One of the tools of visualizing quantitative data. To construct a boxplot it is necessary first to organize the values in increasing order. Next, finding the median of these values we indicate the center. On the last step we find the two medians of these two halves. These points are the first and third quartiles of the data set. Along with the minimum and maximum values they make-up the five point summary. Now, the boxplot consists of a box where these five values are indicated. The form of the boxplot is not standard and could be made both horizontal or vertical. Picture shows one of the possible forms.

boundary value problem For ordinary differential equations. A differential equation along with boundary conditions for solutions. To solve a boundary value problem means to find a solution to the given equation that also satisfies conditions on the boundary. The boundary conditions are chosen to braces Or curly brackets.The symbols { }. One assure that the solution is unique. Example: The of the grouping symbols along with parentheses and

17 brackets. Primarily is used to separate certain numbers and variables to indicate operations to be done first. brachistochrone For given two points on the plane, brachistochrone is the curve of quickest descent between these points. This curve is not the shortest (which is the straight line) and turns out to be a cycloid.

C calculus One of the major branches of mathematics, along with algebra and geometry. Calculus itself consists of two main branches: differential and integral calculus. These two branches are related by the Fundamental Theorem of Calculus.

branches of hyperbola Any non-degenerate hyperbola consists of two separate curves, called Cantor set Also called the Cantor ternary set.Take the closed interval [0, 1] and remove the branches. middle open interval (1/3, 2/3) from if. The result branches of the tangent The graph of the tan- will be the union of two closed intervals [0, 1/3] gent function consists of infinitely many identical and [2/3, 1]. From each of these intervals we again curves, called its branches. These branches are sep- remove their middle third open intervals (1/9, 2/9) arated from each other by the vertical asymptotes and (7/9, 8/9) and get the union of four closed x = π2 + πn, n is an arbitrary integer. intervals [0, 1/9], [2/9, 1/3], [2/3, 7/9], [8/9, 1]. If brackets The symbols [ ]. One of the grouping we continue this process indefinitely, the result will symbols along with parentheses and braces. Primar- be a closed set called Cantor set. This set has many ily is used to separate certain numbers and variables remarkable properties. In particular, it is an infinite set of measure zero and it is a perfect set in the to indicate operations to be done first. sense that for any point in the set there is an infinite sequence of points from that set that approach that point.

cardioid Any of the plane curves given in polar coordinates by the equations r = a(1 ± cos θ) or r = a(1 ± sin θ

18 Cardioids are special cases of lima¸cons. Cartesian coordinate system (1) For points on a plane, also called the rectangular coordinate system. A method of representing points on a plane as ordered pairs of numbers and vice versa. To do so, a pair of perpendicular lines (called axes) are drawn, that intersect at a point called the origin. The horizontal line is called the x-axis and the vertical one – the y-axis. Now, coordinates of any point are determined by drawing perpendicular lines from a point until they reach the axes. The points of these intersections (which represent numbers on corresponding number lines) make up a pair of numbers, called the ordered pair. Conversely, if we have a pair of numbers, then the point corresponding to it could be found by putting these numbers on appropriate axes and drawing two perpendiculars, until they meet at some point.

(3) The system works also for spaces of any dimension n ≥ 2 and even for infinite dimensional spaces. The importance of the Cartesian coordinate system is that it allows to connect geometry with algebra, calculus, and other branches of mathematics. Named after Ren´ e Descartes. Cartesian plane A plane, equipped with the Cartesian coordinate system. Cartesian product For any two given sets A and B (of arbitrary nature) the set of all possible ordered pairs (x, y), where x is an element of A and y is an element of B. The notation is A × B. The Cartesian product can be defined for any number of sets and even for infinitely many sets. Cauchy’s mean value theorem Let the functions f (x) and g(x) be continuous on a closed interval [a, b] and be differentiable on the open interval (a, b). Then there exists a point c, a < c < b, such that f 0 (c) f (b) − f (a) = . g 0 (c) g(b) − g(a) This theorem is the generalization of the Mean value theorem.

(2) For points in three dimensional space. To represent points in the space one additional coordinate axis (called the z-axis) is necessary. This allows to represent any point as an ordered triple, just as in the case of the plane. Similarly, any ordered triple will represent a point in the space.

Cauchy-Euler equation Also called Euler equation. The equation x2 u00 + bxu0 + cu = 0, where b and c are real numbers. The solutions of this equation depend on the solutions of the algebraic equation F (r) = r(r − 1) + ar + b = 0.

19 (1) If the quadratic equation has two real solutions r1 6= r2 then the general solution of the Cauchy-Euler equation is y = c1 |x|r1 + c2 |x|r2 . (2) If the quadratic equation has the a repeated solution r then the general solution is given by the formula y = c1 |x|r + c2 |x|r ln |x|. (3) In the case of two complex solutions r = λ ± iµ, the solution of the differential equation comes in the form y = |x|λ [c1 cos(µ ln |x|) + c2 sin(µ ln |x|)].

the result will be another distribution that is approximately normally distributed and this new distribution has the same mean as the original distribution. The standard deviations are also related. More precisely: Suppose we have some distribution with the mean µ and standard deviation σ. If we form a new distribution from the means of simple random samples of size n, then this distribution will be close to a normal distribution with mean µ and standard deviation σ/n. The larger n is, the better is the approximation.

centroid Let R denote a plate with uniform mass Cauchy-Schwarz inequality For any two nondistribution ρ. The center of mass of this plate is zero sequences of real numbers {ak }nk=1 and {bk }nk=1 , called centroid of R. There are different formulas for the inequality locating the coordinates of the centroid. If the region n n n R is bounded by the function f (x) ≥ 0 and the lines X X X ( ak bk )2 ≤ a2k b2k y = 0, x = a, x = b, then the coordinates of the k=1 k=1 k=1 centroid are given by the formulas Z Z b holds. The equality holds if and only if ak = cbk for 1 b 1 x = xf (x)dx, y = [f (x)]2 dx, some constant c. The same inequality is true even in A a 2A a the case of infinite sequences. Rb Cayley-Hamilton theorem Let A be a square where A = a f (x)dx is the area of the region. In the case when R is bounded by two functions matrix and let f (x) ≥ g(x), the formulas should be adjusted and c0 + c1 λ + c2 λ2 + · · · + cn−1 λn−1 + λn = 0 will include f (x) − g(x) in the formula for x and [f (x)]2 − [g(x)]2 in the formula for y. be its characteristic equation. Then the matrix A chain rule The formula for calculating the derivasatisfies the equation tive of a composite function. (1) For functions of one c0 I + c1 A + c2 A2 + · · · + cn−1 An−1 + An = 0, variable. Let F (x) = f (g(x)) and both f (x) and g(x) are differentiable. Then where I is the identity matrix. F 0 (x) = f 0 (g(x)) · g 0 (x). center of a circle The point that has equal distance from each point of the circle. (2) For functions of several variables: Let z = z(x 1 , · · · , xm ) be a differentiable function of m center of the distribution A somewhat intuitive variables and each of these variables are themnotion that is not exactly defined and may mean difselves differentiable functions of n variables: x1 = ferent thing for different distributions. Most comx (t , · · · , t ), · · · , x = xm (t1 , · · · , tn ). Then the fol1 1 n m monly the center refers to the mean of the distribulowing formulas for the partial derivatives hold: tion µ. An angle, that is formed by two radii

∂z ∂z ∂x1 ∂z ∂xm = + ··· + ∂t1 ∂x1 ∂t1 ∂xm ∂t1

central limit theorem The most fundamental result in Statistics. Informally, it states that if we form a sampling distribution from a large population then

··············· ∂z ∂z ∂x1 ∂z ∂xm = + ··· + ∂tn ∂x1 ∂tn ∂xm ∂tn

central angle of a circle.

20 called the eigenvalues of A. Example: For the matrix change of base For logarithmic function: If a and b are both positive real numbers not equal to 1, and x > 0, then logb x loga x = . logb a

2 A = 1 0

0 3 −4

0 0 1

we have

2 − λ change of variable A method used in many situ0 0 ations to solve certain types of equations or evaluate 3−λ 0 det(A − λI) = 1 0 definite or indefinite integrals. Example: To solve the −4 1 − λ equation x1/2 − 5x1/4 + 6 = 0 we make a change of variable y = x1/4 to translate to the quadratic equa= (2 − λ)(3 − λ)(1 − λ), tion y 2 − 5y + 6 = 0 which has the solutions y = 2, 3. Returning to the original variable x we get x1/4 = 2, 3 which has three real and distinct roots: 1,2,3. See or x = 16, 81. For examples with integration see the entry substitu- also eigenvalues and eigenvectors. (2) For a linear homogeneous differential equation tion method. characteristic equation (1) The equation p(x) = 0, where p is the characteristic polynomial of a given square matrix. If the matrix has the size n × n, then the polynomial has degree n and, by the Fundamental Theorem of Algebra, has exactly n roots (if counted with multiplicities). The roots of the equation may be both real or complex and some of them might be repeated. See also eigenvalues, eigenvectors, and eigenspace. (2) For a linear homogeneous differential equation a0 y (n) + a1 y (n−1) + · · · + cn−1 y 0 + cn y = 0 the corresponding polynomial equation a0 rn + a1 rn−1 + · · · + cn−1 r + cn = 0. The solutions of this equation are called characteristic roots. For the importance and the use of these roots see the article linear ordinary differential equations.

a0 y (n) + a1 y (n−1) + · · · + cn−1 y 0 + cn y = 0 the corresponding polynomial p(r) = a0 rn + a1 rn−1 + · · · + cn−1 r + cn of degree n with respect to the variable r is its characteristic polynomial. The roots (zeros) of the polynomial are the characteristic roots of the polynomial. Chebyshev equation

The differential equation

(1 − x2 )y 00 − xy 0 + αy = 0. Solutions of this equation when α is a non-negative integer n, are called Chebyshev polynomials. These polynomials could be written in many different forms. The simplest and most used form is

√ √ (x + x2 − 1)n + (x + x2 − 1)−n characteristic polynomial (1) Let A be an n × Tn (x) = . 2 n matrix and I is the identity matrix of the same size. The determinant of the matrix A − λI is a polynomial, called the characteristic polynomial of the matrix A. This polynomial has degree n and, by the Fundamental Theorem of Algebra, has exactly n chi-square distribution The distribution of roots (if counted with multiplicities). This roots are

21 for any combination of indices 1 ≤ i, j ≤ n. closed curve A curve that has no end points. If a plane curve is given by the parametric equations x = x(t), y = y(t), t ∈ [a, b], then the curve is closed if x(a) = x(b) and y(a) = y(b). closed interval Intervals of the form [a, b], where both end-points are finite and included.

variance. There are multiple definitions for this distribution and the following is one of the simplest: χ2 =

(n − 1)s2 , σ2

where n is the sample size, s2 is the sample variance and σ 2 is the population variance. Because variance is positive the curve of the distribution is always on the right side. As the graph shows, the distribution becomes more symmetric with the increasing size of the sample.

closed set A set that contains all of its boundary points. For example, the interval [0, 1] is a closed set because it contains both of its boundary points 0 and 1, while the interval (0, 1) is not closed. The set {(x, y)|x2 + y 2 ≤ 1} is a closed set on the plane but the set {(x, y)|x2 +y 2 < 1} is not a closed set because it does not contain its boundary. closed surface In the simplest form, a closed surface is the boundary of a bounded solid domain. The general definition, given in advanced calculus courses, requires notions from set theory and topology.

closed under A set of numbers (or, more generally, some objects) is closed under certain operation, if the result of that operation also belongs to that set. Examples. (1) The set of all integers is closed chord For any given curve, a straight line segment, under both operations of addition and multiplication, that connects any two points on the curve. Most because the result is also an integer. (2) The set of commonly is used for the circle. real numbers is closed under addition, subtraction, multiplication, and division (except by zero), but it circle A plane geometric figure with the property is not closed under the operation of square root exthat there exists a point (called center of the circle) traction. (3) The set of all square matrices is closed and a positive number r (called the radius), such under addition, scalar multiplication and also matrix that the distance from each point of that figure to multiplication. the center is equal to r. coefficient A word most commonly used to indicircular cylinder A cylinder with a circle base. circumference

The perimeter of a circle.

cate the non-variable factor of a term of a polynomial. See coefficients of a polynomial. This term is also used to describe the numeric components in functions other than polynomials.

Clairaut’s theorem Also known as the theorem coefficient matrix For a system of linear equaof the change of order of partial differentiation. If tions the function f (x1 , x2 , . . . , xn ) has continuous second a11 x1 + a12 x2 + · · · + a1n xn = b1 partial derivatives at some point (a1 , a2 , ..., an ) in its a21 x1 + a22 x2 + · · · + a2n xn = b2 domain, then ∂2f ∂2f (a1 , a2 , ..., an ) = (a1 , a2 , ..., an ) ∂xi ∂xj ∂xj ∂xi

··························· am1 x1 + am2 x2 + · · · + amn xn = bm

22 π π the m × n matrix that consists of all coefficients of cot( − θ) = tan θ, tan( − θ) = cot θ; 2 2 the system (but not the constant parts on the right π π sides): csc( − θ) = sec θ, sec( − θ) = csc θ, 2 2 a11 a12 . . . a1n which are true for any real values of θ where the func a21 a22 . . . a2n . . tions are defined. . . .. .. .. column matrix A matrix, that consists of one colam1 am2 . . . amn umn, or an m × 1 matrix. It is the same as a vector, When a column of the constant terms is attached, written in column form. Example: the matrix is called augmented. 3 coefficients of a polynomial In the polynomial −4 . of arbitrary degree 1 p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , column space Let A be an m × n matrix. The the constants a0 , a1 , · · · , an . subspace of the space Rm that is spanned by the coefficients of power series In the power series column-vectors of that matrix is called the column space of A. ∞ X n an (x − c) column vector A vector written in column form. n=0 The same as column matrix above. the numeric constants an , n = 0, 1, 2, 3 · · ·. combination For the given set S of n elements, cofactor For a given n × n matrix A, the (i, j)th cofactor is the determinant of the (n−1)×(n−1) matrix obtained by deleting the ith row and jth column of A, multiplied by (−1)i+j . Example: If −4 4 1 A = 0 −1 3 2 5 2 then the cofactor corresponding to the element a23 (which is 3) is −4 4 (−1)2+3 det = (−1)(−20 − 8) = 28. 2 5

combination of k elements is a subset of S, that contains exactly k elements. At least one element should be different and two sets with the same elements are considered the same. For example, the subsets {1, 2, 3} and {2, 1, 3} are the same. The number of combinations of k elements out of n is given by the formula n n! = Ckn = , k k!(n − k)! where 0 ≤ k ≤ n. See also binomial coefficients and permutation.

common denominator If two or more fractions have the same denominator, then that denominator is called the common denominator (of the fractions). See also minors of a matrix. If the denominators are different, the fractions could cofactor matrix Another name for the adjoint be changed to an equivalent form with a common matrix. denominator. See least common denominator for decofunction The common name of trigonometric tails. functions cos x, cot x, csc x. common factors In any factorization of two or cofunction identities In trigonometry, the identities π π cos( − θ) = sin θ, sin( − θ) = cos θ; 2 2

more natural numbers, any factors that are the same for all of these numbers. Example: The numbers 24 = 23 · 3 and 30 = 2 · 3 · 5 have three common factors, 2, 3, and 6.

23 common logarithms Logarithmic function with complementary angles Two angles are complebase 10. The base of common logarithms is often mentary, if the sum of their measures is 90◦ (in denot written but is understood: log x = log10 x. gree measure) or π/2 (in radian measure). common multiple For two or more integers, any integer that is divisible by all of them. The number 30 is a common multiple of 10 and 6, because 30 = 10 · 3 and 30 = 6 · 5.

completing the square If a quadratic trinomial is not a perfect square, it is still possible to represent it as a sum of a perfect square and some constant. This procedure is called completing the square. If the trinomial is ax2 + bx + c, then we isolate the first commutative property for addition For any two terms and write given real or complex numbers a and b, a + b = b + a, b or, the order of the addends does not change the sum. ax2 + bx + c = a x2 + x + c a commutative property for multiplication For any given real or complex numbers a and b, a·b = b·a, and notice that in order to make the expression inside or, the order of the factors does not change the prod- the parenthesis a perfect square we are missing the uct. square of half of the linear term’s coefficient. Adding commuting matrices Two square matrices A and and subtracting that number we get B such that A · B = B · A. b2 b2 b 2 a x + x+ 2 − 2 +c comparison properties of the integral For ina 4a 4a tegrable functions f (x) and g(x) of one real variable b b2 b2 2 defined on [a, b]: = a x + x + +c − Rb 2 a 4a 4a (1) If f (x) ≥ 0 then a f (x)dx ≥ 0 Rb Rb 2 (2) If f (x) ≥ g(x) then a f (x)dx ≥ a g(x)dx. b b2 Rb =a x+ − + c. 2a 4a (3) If m ≤ f (x) ≤ M , then m(b − a) ≤ a f (x)dx ≤ M (b − a). Examples: comparison theorem for integrals Let f and g be two functions defined on [a, ∞), with the property f (x) ≥ Rg(x) for x ≥ a. R∞ ∞ (1) If a f (x)dx is convergent, then a g(x)dx is also convergent. R∞ R∞ (2) If a g(x)dx is divergent, then a f (x)dx is also divergent.

(1) x2 − 6x + 7 = (x2 − 6x + 9) − 9 + 7 = (x − 3)2 − 2. (2) 3x2 + 7x − 5 = 3(x2 + 37 x) − 5 = 3(x2 + 73 x + 49 36 − 7 49 49 7 2 109 49 2 ) − 5 = 3(x + x + ) − − 5 = 3(x + ) − 36 3 36 12 6 12 .

complex conjugates Complex numbers with the same real parts and opposite imaginary parts: a + ib and a−ib. The complex conjugate of the number z is denoted by z. A very important property of complex comparison test for series Let {an } and {bn } be conjugates is that their product is always a positive two numeric sequences. number or zero (if the number is zero itself): P∞ b is conver(1) If 0 ≤ aP n ≤ bn for all n and n n=1 ∞ (a + ib)(a − ib) = a2 − i2 b2 = a2 + b2 ≥ 0. gent, then n=1 an is convergent. P∞ (2) If P an ≥ bn ≥ 0 for all n and n=1 bn is divergent, ∞ This fact is used in the process of the division of two then n=1 an is divergent. complex numbers. complement of a set Let S be a space of sets or a set of some elements and assume A is an element of complex exponents The complex exponent for ¯ that the base e is defined by the use of Euler’s formula. If that space. The complement of A is the set A, has no intersection with A (A ∩ A¯ = ∅), and their z = x + iy is some complex number, then union ”fills” the space, A ∪ A¯ = S. ez = ex (cos y + i sin y).

24 For any other positive base a 6= 1 the complex ex- vector-functions also. ponent could be defined similarly with the use of the components of a vector Let v be some vecidentity a = eln a . tor in a vector space V . Then it could be writcomplex fraction A fraction, where the numera- ten in its component form v = (v1 , v2 , · · · , vn ) and tor, or denominator or both are fractions themselves. the scalars v1 , v2 , · · · , vn are the components of this Examples: vector. Components depend on the basis of the x 2 space V and vary with change of basis. Exam2x + 1 . 3 , ple: the vector v = (2, 3) in the standard basis 7 x2 − 1 e1 = (1, 0), e2 = (0, 1) of R2 , in the nonstandard 13 x−5 basis e01 = (−1, 0), e02 = (0, −1) will have compocomplex numbers Numbers of the form z√= x + nents (−2, −3). iy, where x and y are real numbers and i = −1, is the imaginary unit. x is called the real part of the composite function See composition of functions. complex number and y is the imaginary part. Two composite number A natural number, that could complex numbers are equal, if their real and imagi- be written as a product of two other natural numbers nary parts are identical: a + ib = c + id if and only if different from 1 or the number itself. If it is not posa = c and b = d. Each complex number can be rep- sible, the number is called prime. resented graphically as a point on the plane, where the real part is identified with the x-coordinate and composition of functions (1) For functions of one variable. Let f (x) and g(x) be two functions the imaginary part with the y-coordinate. Complex numbers also have a trigonometric repre- and assume that the range of g is contained in the sentation ofpthe form z = r(cos θ + i sin θ), where domain of f . Then the composition of f with g r = |z| = x2 + y 2 is the modulus of the number is defined to be the function h(x) = f (g(x)). The and the angle θ, (called the argument of z) is deter- notation h = f ◦ g is commonly used. The operation of composition is not commutative. Exammined from the equation tan θ = y/x. 2 For operations with complex numbers see addition ple: Let f (x) = x + 1 and g(x) = 2x − 1. Then 2 2 and subtraction of complex numbers, division of com- (f ◦ g)(x) = 4x − 4x + 1 and (g ◦ f )(x) = 2x + 1. If plex numbers, multiplication of complex numbers, (2) For functions of several variables. f = f (x1 , · · · , xm ) and there are m functions DeMoivre’s theorem. gk (x1 , · · · , xn ), 1 ≤ k ≤ m, of n variables, then the complex plane A plane equivalent to the Carte- composition function can be defined by the formula sian plane with one significant difference: the points h(x1 , · · · , xn ) = on the plane are not viewed as ordered pairs (x, y), but rather as points with real and imaginary parts f (g1 (x1 , · · · , xn ), · · · , gm (x1 , · · · , xn )). coinciding with these same values x, y. Hence, arithmetic operations can be done with the points on the In this definition we have to assume also that the complex plane according to the rules of operations combined ranges of the functions gk are in the domain of f . In the several variable case the composiwith complex numbers. tion is also not commutative. component function If r is a vector-function in the three dimensional space, then it could be written composition of linear transformations Let T1 : V1 → V2 be a linear transformation between two linin the form ear vector spaces V1 and V2 . Assume also, that another linear transformation T2 between V2 and some r(t) = (f (t), g(t), h(t)). other linear vector space V3 is defined in such a way, The functions f, g, h are called component func- that its domain is contained in the range of the transtions. The same notion is valid for two dimensional formation T1 . Then the composition transformation

25 x2 is positive”, ”If a function is differentiable, then it is continuous”. To each conditional statement A ⇒ B, there exist three other statements. (1) Converse: B ⇒ A, meaning ”If B, then A”. If the direct statement is true, the converse may or compound interest When a given amount of may not be true. money is invested and the interest is paid not only on (2) Inverse: not A ⇒ not B, meaning ”If A is not the principal amount but also on the interest earned true, then B is not true”. As in the case of converse, on the principal, it is called compound interest. If the inverse may or may not be true. the principal amount is denoted by P , the interest (3) Contrapositive: not B ⇒ not A, meaning ”If rate by r and interest payments are made n times a B is not true, then A is not true”. Direct and year, then the formula for the amount A after t years contrapositive statements are equivalent in the sense is that either they both are true or both are false. r nt . A=P 1+ See also biconditional statement, logical contraposin tive. See also continuously compounded interest. concave function A differentiable function on conditionally convergent series A series that absolute some interval is concave down if its derivative is a de- converges but the series formed by the P∞ (−1)n creasing function. Geometrically this property means values of its terms does not. The series n=1 n that the tangent line to the graph of the function is conditionally convergent because the series formed its absolute values is the divergent harmonic series at any point is above the graph. Similarly, a dif- by P∞ n=1 1/n. If a series is conditionally convergent, ferentiable function on some interval is concave up if its derivative is an increasing function. Geomet- then its terms could be rearranged so that the sum rically this property means that the tangent line to equals any given number.

T (v) = T2 (T1 (v)) = (T2 ◦ T1 )(v) is a transformation between the spaces V1 and V3 . The composition operation is not commutative: T1 ◦ T2 6= T2 ◦ T1 in general.

the graph of the function at any point is below the graph. Concave up functions are also called convex. concavity

The property of being concave.

concavity test Let f (x) be a twice differentiable function on some interval I. (1) If f 0 (x) > 0 for all values of x on the interval, then the function is concave up. (2) If f 0 (x) < 0 for all values of x on the interval, then the function is concave down. conditional probability The probability of some event A under the assumption that event B happens too. The notation is P (A|B). If two events A, B are not mutually exclusive, then the probability that both events happen is given by the general multiplication formula P (A ∩ B) = P (A)P (B|A). cone A three dimensional geometric object generated by a line (called the generator) rotating about a conditional statement A logical statement of fixed point on the line called the apex. Both the surthe form ”If A, then B”, written also in the short face and the solid generated this way are called cone. form A ⇒ B. Examples include: ”If x is real, then The picture above shows one half of a right circular

26 cone. confidence interval In statistics. An interval of real values where we expect to have the ”majority” of values of some distribution concentrated. The two endpoints of that interval are given by the expression Estimate ± M argin of Error. For example, for the distribution of proportions, this interval could be given by the expression pˆ ± z ∗ SE(ˆ p), where pˆ is the estimated proportion of the sample data, SE(ˆ p) is the standard error of that sample data, and z ∗ is the critical value corresponding to the given confidence level. The length of the interval depends on the size of the data set and on the desired confidence level.

etc. conjugates The expressions a + b and a − b are called conjugates or conjugate pairs. The term most often refers to complex conjugates. It is√also √ used for √ √ expressions of the form a + b and a − b. Using these conjugates is important when rationalizing denominators containing these expressions. conjugate axis

See hyperbola.

connected region If any two points of a region can be connected by a path (curve) that lies completely inside that region, then the region is called connected.

conservative vector field A vector field F is called conservative, if there exists a scalar function f such that its gradient vector field coincides with confidence level A percentage we chose to be con- F : F=5f . fident of the accuracy of statistical estimates that consistent linear system A system of equations come from sample data. The most common values which has at least one solution. When the system are 90%, 95%, and 99%. The confidence level is ashas no solution, it is called inconsistent. sociated with critical values. constant A quantity that does not change, usually congruence An equivalence relation between two some type of number. or more sets. Geometric figures are called congruent if one figure (plane or solid) could be moved and/or constant multiple rule See differentiation rules. rotated and/or reflected to coincide with the other constraint A term used as a synonym of restricfigure. Congruence is different from equality in the tion. The constraints could be on an independent sense that these two figures are not the same because variable as well as on a function itself. Example: originally they occupied different places on the plane Find the maximum value of the function f (x, y) = or in space while equal figures are supposed to have 2x2 − 3xy + y 2 under the constraint x2 + y 2 = 1. all components of the same size and occupy the same continued fraction An expression of the form space. conic section The result of cutting a double cone by a plane. Depending on the position of the plane the result could be a circle, ellipse, parabola, hyperbola or a degenerate conic section: a point, a pair of intersecting lines, or the empty set. An alternative geometric definition could be given using eccentricity. Algebraically, any conic section could be described as a solution of a quadratic equation in two variables: Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0, where A, B, C, D, E, F are real constants. See the entries ellipse, hyperbola, parabola for the details and additional information regarding foci, directrix, axes

b1

a0 + a1 +

b2 a2 + · · ·

.

Continued fraction might be finite or continue indefinitely. continued fraction expansion Representation of a number or a function as a finite or infinite continued fraction. Example: √

1

2=1+ 2+

1 2 + ···

.

27 person taking them. This method is not considered continuity of a function The function f (x), de- scientifically reliable or unbiased. fined on some interval [a, b] is continuous at a point convergence For specific definitions see converc ∈ (a, b), if gent integrals, convergent sequences, convergent selim f (x) = f (c). x→c ries. See also absolutely convergent series, conditioncontinuity on an interval The function is contin- ally convergent series. uous on an interval, if it is continuous at each point convergent integral A definite integral with finite of that interval. If the function is defined on a closed value. For proper integrals this means that the Rieinterval [a, b] then for the continuity at the end-points mann sums have finite limit. The same applies also a and b see the entries continuity from the right and to improper integrals. continuity from the left respectively below. convergent sequence A numeric sequence that continuity from the left The function f (x) is approaches a finite value. The formal definition is: continuous from the left at some point c, if at that Let {an }∞ n=1 be a sequence of (real or complex) numpoint the left-hand limit exists and is equal to the bers. The sequence converges to some number A, if value of the function at that point: limx→c− f (x) = for any given ε > 0 there exists an N > 0, such that f (c). |an − A| < ε whenever n > N . In an equivalent nocontinuity from the right The function f (x) is tation, limn→∞ an = A. continuous from the right at some point c, if at that Example: The sequence {1 + 1/n} converges to 1 as point the right-hand limit exists and is equal to the n → ∞. value of the function at that point: limx→c+ f (x) = convergent series A numeric series convergent Pis f (c). n if the partial sums of that series Sn = k=1 ak form a continuous function See continuity of a function. convergent sequence. More precisely, if for any given ε > 0 there exists a number N such that |Sn − L| < ε continuously compounded interest When a for any n > N , we say that the series converges to L. given amount of money is invested and the interest is Example: The series paid not only on principal amount but also on interest ∞ X earned on principal, and the number of payments is 1 unlimited, then it is called continuously compounded p n n=1 interest. If the principal amount is P and the interest rate is r, then the amount of money after t years is is convergent for any value p > 1. The same exact given by the formula A = P ert . See also compound definition is valid also for functional series, such as power series or Fourier series. interest. contraction operator Or contraction transformation. An operator applied to a vector does not change its direction but makes its magnitude smaller. The operator is given by the formula T x = kx, where 0 ≤ k ≤ 1. In the case k ≥ 1 the transformation is called a dilation operator. For a more general case when not only the magnitude but also the direction is changed, see expansion operator.

converging solutions A term commonly used to describe approximate solutions that have the property of approaching the exact solution during a certain limiting process. As an example see Newton’s method for conditions when approximate solutions approach exact solution. converse

See conditional statement.

convex function A differentiable function on some interval is convex, if its derivative is an increasing convenience sampling A type of sampling when function. Geometrically this property means that the the samples are taken based on the convenience of the tangent line to the graph of the function at every contrapositive

See conditional statement.

28 point is always below the graph. Convex functions also are called concave up.

clidean space Rn , that lie on the same plane.

correlation coefficient Suppose we have data in convex set A set in Euclidean space with the prop- the form of ordered pairs (x, y) and there is some erty that for any two points points in that set, the kind of relationship between x and y-variables. Corline connecting those points lies completely in the set. relation coefficient measures the strength of that relationship in numerical terms. To calculate that coconvolution integral For two functions defined on efficient, usually denoted by r, we first normalize the the real axis R, the formal integral x and y-values by finding their z-scores denoted by Z ∞ zx and zy respectively. The sum f (t)g(x − t)dt. (f ∗ g)(x) = P −∞ zx zy r = Convolution is commutative in the sense that f ∗ g = n−1 g ∗ f . Another important property of convolution is that the Laplace transform translates convolu- is the correlation coefficient. It is defined in a way tion into multiplication: Let Lf (x) and Lg(x) be that −1 ≤ r ≤ 1. A positive r indicates positive Laplace transforms of f and g respectively. Then correlation and a negative r indicates negative correlation. The closer r is to 1 or -1, the stronger is L(f ∗ g)(x) = Lf (x) · Lg(x). the linear relationship between the x and y-values. coordinate axes See Cartesian coordinate system. coordinate matrix If S={v1 , v2 , ..., vn } is a basis in some vector space V then any vector v from V could be written as v = c1 v1 + c2 v2 + · · · + cn vn . The n × 1 column-matrix that consists of the scalars c1 , c2 , ..., cn is called the coordinate matrix of the vector v with respect to the basis S. coordinate plane The plane that is determined by any pair of coordinate axes. See Cartesian coordinate system. coordinate vector The same as coordinate matrix, corresponding angles Corresponding angles are formed when a line crosses two coplanar lines. The only written in the form of a 1 × n row-matrix. corresponding angles are not necessarily congruent, coordinate system Any of the methods to idenbut they are if the coplanar lines are also parallel. tify points on the plane or in space with an ordered In the figure above the lines a and b are parallel and pair or an ordered triple. See the coordinate systems the line t is the transversal. Congruent angles are Cartesian, cylindrical, polar, rectangular, spherical. marked accordingly. coordinates In any coordinate system the ordered pairs or triples corresponding to a point. The same cosecant function One of the six trigonometric point may have different coordinates in different cofunctions. Geometrically, the cosecant of an angle in ordinate systems. Example: The point with coordi√ a right triangle is the ratio of the hypotenuse of the nates (−2, 2 3) in the Cartesian system has coorditriangle to the opposite side. The cosecant could also nates (4, 2π/3) in the polar system. be defined as the reciprocal of the sine function. The coplanar vectors Two or more vectors in the Eu- function csc x could be extended to all real values ex-

29 actly as the sin x function is extended. The domain of csc x is all real values, except x = πn, n any integer, and the range is (−∞, −1] ∪ [1, ∞). csc x is 2π-periodic. The cosecant function is related to other trigonometric functions by many identities, the most important of these are csc x = 1/ sin x, 1 + cot2 x = csc2 x. The derivative and integral of this function are given by the formulas d csc x = − csc x cot x, dx Z csc xdx = ln | csc x − cot x| + C.

cosine function One of the six trigonometric functions. Geometrically, the cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse of the triangle. A more general approach to extend the cos x function for any real number x is as follows: Let P = (a, b) be any point on the plane other than the origin and θ is the angle formed by the x-axes and the terminal side, connecting the origin and P . Then cos θ = √a2a+b2 . Next, after establishing one-to-one correspondence between angles and real numbers, we can have the cosine function defined for all real numbers. The range of cos x is [−1, 1] and it is 2π-periodic. The cosine function is related to other trigonometric functions by various identities. The most important is the Pythagorean identity sin2 x + cos2 x = 1. The derivative and indefinite integral of this function are: Z d (cos x) = − sin x, cos xdx = sin x + C. dx

cost function The total cost to a company to produce x units of certain product. Usually denoted by C(x). The derivative of this function is called the marginal cost function. See also average cost function, revenue function..

cotangent function One of the six trigonometric functions. Geometrically, the cotangent of an angle in a right triangle is the ratio of the adjacent side of the triangle to the opposite side. It could also be defined as the reciprocal of the tangent function. The function cot x could be extended as a function of all real numbers except for x = πn, n any integer, and the range is all of R. The cot x function is π-periodic. The cotangent function is related to the other trigonometric functions by various identities. The most important of these are the identities cot x = cos x/ sin x, cot x = 1/ tan x and a version of the Pythagorean identity 1 + cot2 x = csc2 x. The derivative and integral of this function are given by the

30 formulas d cot x = −csc2 x, dx

Z cot xdx = ln | sin x| + C.

Calculating these determinants, we get |A| = 44, |A1 | = −40, |A2 | = 72, |A3 | = 152. Now, Cramer’s rule gives the solution (−12/11, 18/11, 38/11).

coterminal angles Two angles in standard position which have the same terminal side (see the entry angle for explanation of terms above). Any two coterminal angles differ in size by an integer multiple of 360◦ (in degree measure) or 2π (in radian measure).

critical point Also called critical number or value. For a continuous function f (x) on some interval I, a point c ∈ I is critical, if f 0 (c) = 0 or f 0 (c) does not exist. Critical points are important because if a function has a local extremum at some point in its domain, then that point is a critical point. The opposite counting numbers Another name for natural is not always true. Examples: (1) For the function f (x) = x − 2 sin x on [0, 2π] the critical points are numbers. x = π/3, x = 5π/3. The first one is the local miniCramer’s rule One of the methods of solving sys- mum and the second one the local maximum of the tems of linear algebraic equations which involves the function on the given interval. (2) For the function use of determinants. For the system of n equations with n unknowns n 2x if x ≥ 0 f (x) = x if x < 0 a11 x1 + a12 x2 + · · · + a1n xn = b1 a21 x1 + a22 x2 + · · · + a2n xn = b2 ······························ an1 x1 + an2 x2 + · · · + ann xn = bn the solution is given by the formulas xk = det(Ak )/ det(A), 1 ≤ k ≤ n. Here A is the matrix of the given system of equations and the matrices Ak are found from A by removing the kth column and substituting it by the column of constants [b1 , b2 , · · · , bn ]. This formula works if and only if the determinant det(A) 6= 0. Example: Solve the system x1 +

+xn = 6

−3x1 + 4x2 + 6xn = 30 −x1 − 2x2 + 3xn = 8 The four determinants for this system are: 1 6 0 2 0 2 |A| = −3 4 6 , |A1 | = 30 4 6 −1 −2 3 8 −2 3 1 1 6 2 0 6 |A2 | = −3 30 6 , |A3 | = −3 4 30 −1 8 3 −1 −2 8

the only critical point is x = 0 but at that point the function has neither a minimum nor a maximum value. The notion of a critical point is also extended to functions of several variables. A point (c1 , c2 , c3 ) is a critical point of a function f (x, y, z) of three variables if either all first partial derivatives are equal to zero or some of the first partial derivatives do not exist at that point. As in the case of the functions of one variable, here also the local maximum and minimum values are possible at critical points only, but the opposite is not true: not all critical points are local minimum or maximum points for the function. These kind of points are called saddle points. critical value In statistics. A numeric value associated with any distribution that depends on the type of distribution and the confidence level desired. For the standard normal distribution the critical values corresponding to the most common confidence levels 90%, 95%, 99% are z ∗ = 1.165, 1.96, 2.576 respectively. cross product For two vectors in three dimensional space only. Let u = (u1 , u2 , u3 ) and v = (v1 , v2 , v3 ) be vectors in R3 . Then their cross product is defined to be another vector from the same space,

31 given by the formula u × v = (u2 v3 − u3 v2 , u3 v1 − u1 v3 , u1 v2 − u2 v1 ). This vector could be expressed as a determinant: i j k u × v = u1 u2 u3 . v1 v2 v3

To find the solutions of equation (2), we need to make the back substitution x = y − a/3. The solutions of equation (1) are obviously the same as that of equation (2). Note also, that at least one of the solutions of the equations (1)-(3) is always real. Example: For the equation x3 − 8x − 3 = 0 the solu√ √ −3+ 5 −3− 5 tions are 3, 2 , 2 .

cubic polynomial A polynomial of the third de3 2 The cross product is not commutative: u × v = gree p(x) = ax + bx + cx + d. In the particular case, the function f (x) = ax3 is called the cubic function. −v × u. cubic equation An equation of the form p(x) = 0, where p is a cubic polynomial. By the Fundamental Theorem of Algebra, any cubic equation has exactly three solutions counting multiplicities of zeros. Similar to quadratic equations, any cubic equation could be solved by a formula involving radicals, however this formula is very complicated and difficult to use. Here is a short demonstration of that formula and its deduction. To solve the general cubic equation αx3 + βx2 + γx + δ = 0,

α 6= 0, (1)

cubic root Formally, the inverse of the cubic function. A given number a is the cubic root √ of another number, b, if a3 = b. The notation is a = 3 b. curl of a vector field Let F = P i + Qj + Rk be a vector field in R3 and assume that all partial derivatives of the components P, Q, R exist. Then the vector function ∂R ∂Q curlF = − i ∂y ∂z ∂P ∂R ∂Q ∂P + − j+ − k ∂z ∂x ∂x ∂y

we divide it by α and get a monic equation (equation is the curl of the vector field. See also divergence of with leading coefficient equal to 1) a vector field and gradient vector field. x3 + ax2 + bx + c = 0. (2) curvature Let C be a smooth curve defined by some vector function r = r(t) and let s = s(t) be that Denoting x = y − a/3 we reduce this equation to a curve’s arc length. If T = r0 /|r0 | is the unit tangent simpler equation of the form vector to the curve then the curvature of the curve is y 3 + py + q = 0, (3) given by the formula dT where p = b − a2 /3, q = c − ab/3 + 2a3 /27. Let . κ= r denote one √ of two complex cubic roots of unity: ds √ r =p −1/2 + 3i/2 or r = −1/2 − √3i/2 and let 2 3 ∆ = −(27q √ + 4p ), A = (−27q + 3i 3∆)/2, B = curve (1) A plane curve is a set of all ordered (−27q − 3i 3∆)/2. The choice of the two possible pairs (f (t), g(t)), where f and g are continuous square roots for ∆ and three possible cubic roots of A functions defined on some interval I = [α, β]. If no 1/3 and B must be made so that (AB) = −3p. With points repeat (i.e., if (f (t1 ), g(t1 )) 6= (f (t2 ), g(t2 )) this notations, the three roots of equation (3) are for two values t1 6= t2 ), then the curve is called given by the following formulas: simple. If (f (α), g(α)) = (f (β), g(β)), then we 1/3 1/3 have a closed curve. If the functions defining the A +B y1 = , y2 = rA1/3 + r2 B 1/3 , curve are differentiable, the curve is called smooth. 3 Accordingly, if they are piece-wise differentiable then y3 = r2 A1/3 + rB 1/3 . the curve is called piece-wise smooth. A plane curve

32 could be also given by a polar equation. See also boundary curve. (2) A curve in R3 is a set of all ordered triples (f (t), g(t), h(t)) with all functions continuous on some interval I. The definitions of closed, simple, and smooth curves remain the same as above. For calculations of lengths of curves see length of parametric curve, length of polar curve, length of a space curve.

cycloid A plane curve that appears as a trace of a point on a circle, when the circle rolls on a straight line. The parametric equations for the cycloid are given by

cylinder Geometrically, a cylinder is a three dimensional surface that consists of lines that are parallel to some given line and pass through some closed plane curve. The most commonly used cylinders are right circular cylinders that pass through a circle on the plane and are perpendicular to that plane. Algebraically, a right circular cylinder is given by the formula x2 + y 2 = r2 , where the third variable z is free to assume any real values. Another example of a cylinder is a right elliptic cylinder given by the formula of ellipse x2 /a2 + y 2 /b2 = 1 and the variable z is again free. cylindrical coordinates A coordinate system in three dimensional space that uses polar coordinates in two dimensions and the rectangular coordinate in the third dimension. The relations are given by the formulas x = r cos θ, y = r sin θ, z = z.

x = a(θ − sin θ), y = a(1 − cos θ),

Here r2 = x2 + y 2 and the angle is determined from the equation tan θ = y/x.

where a is the radius of the circle and θ is the angle formed by two radii in the circle, one connecting the center with the tracing point on the cycloid and the other connecting the center with the point where the circle touches line.

cylindrical shell A geometric solid which is the difference of two concentric circular cylinders with the same axis but different radii. For applications of cylindrical shells see volume.

33

D

decomposition of matrices Writing a given matrix as a sum or product of two or more matrices. For a specific way of splitting a matrix see LU(lower upper)-decomposition.

damped vibration If an object oscillates and another force (such as friction) affects its motion by decreasing its amplitude, then this results in a damped vibration of the object. This kind of motion is described by the differential equation

decreasing function The function f (x) defined on some interval I is decreasing, if for any two points x1 , x2 ∈ I, x1 < x2 , we have f (x1 ) > f (x2 ). The function f (x) = 2−x is an example of a decreasing function.

my 00 + cy 0 + ky = 0,

decreasing sequence A sequence of real numbers {an }, n ≥ 1, is decreasing, if am < ak whenever m > k. The sequence an = 1/n2 is an example of a decreasing sequence.

where c is the damping constant.

data A set of values collected in the process of sampling or a census. The values of the data could be definite integral (1) Let the function f (x) be denumerical (quantitative) or categorical (qualitative). fined and continuous on some finite interval [a, b]. Definite integral of a function could be defined in data analysis A collection of statistical methods many different but equivalent ways and we present designed to evaluate and draw conclusions from a some of this definitions. body of data. For specific methods see, e.g. conLet us divide the interval into n equal parts of size fidence interval, hypothesis testing, analysis of vari∆x = (b − a)/n. Denote the endpoints of these parts ance, etc. by x0 (= a), x1 , x2 , · · · , xn (= b) and chose arbitrary decimal numbers The numeric system with base points x∗1 , x∗2 , · · · , x∗n in each of these smaller interten. This system uses ten digits 0,1,2,3,4,5,6,7,8,9 vals [xi−1 , xi ], 1 ≤ i ≤ n. Then the definite integral and the place value system, which takes into consid- of f on the interval [a, b] is the limit eration the position of the digit along with its value, Z b n X to write all real numbers. For example, in the number f (x)dx = lim f (x∗i )∆x n→∞ 3531 the two 3’s have different values because of being a i=1 in different positions. The first 3 represents 3000 and the second one 30. To represent fractional and even if the limit exists. In special cases of this definition irrational numbers in decimal system, we use the dec- the sample points x∗i can be chosen to be the left imal point (in many countries a coma is used), which endpoints or right endpoints, or the midpoints of inseparates the whole part of the number from the frac- tervals. They all are equivalent to the more general tional. In the number 43.587 the 43 is the whole definition. part and the digits after the point represent the frac- The sum in the definition of the integral is called a tional part. Hence, 43.587 = 43+0.587. Any fraction Riemann sum and the definite integral is called the (rational number), could be written as either a ter- Riemann integral. The calculations for definite inminating or non-terminating, repeating decimal. On tegrals are done primarily by the use of the Fundathe other hand, any irrational number can be writ- mental Theorem of Calculus, rather than using the ten as an infinite non-repeating√decimal. Examples: definition directly. Examples: 3/8 = 0.375, 1/3 = 0.333..., 2 = 1.414213562.... Z 4 See also binary numbers. 4x3 dx = x4 |41 = 256 − 1 = 255, 1 deciphering matrix The matrix used to decipher Z π a coded message, usually, the inverse of the matrix sin xdx = − cos x|π0 = −(−1) − (−1) = 2. that coded the message. 0

34 For a positive function f (x) ≥ 0 on [a, b] the definite integral is just the area under the graph of this function. More generally, the definite integral is the difference of the areas above and below the x-axis for the graph of a given function. (2) This definition could be used to include also functions with finite number of jump discontinuities. The notion of the definite (Riemann) integral could be generalized to include more general functions, such as with infinite discontinuities or defined on infinite intervals. See improper integral. (3) Further generalizations of the definite integral resulted in development of other types of integrals (Stiljes, Lebesgue, Denjoy, etc.).

the measure 1◦ . Hence, the right angle would have 90◦ and the straight angle 180◦ . See also angle. demand function Also called price function. The price of the product when a company wants to sell x units of that product, usually denoted by p(x). The graph of a demand function is called a demand curve. Closely related are revenue function, profit function. DeMoivre’s theorem Let the complex number z be written in trigonometric form: z = reiθ = r(cos θ + i sin θ). Then for any integer n z n = rn einθ = rn (cos nθ + i sin nθ). denominator (1) For the numeric fraction ab , the number b. (2) For the rational function p(x) q(x) , the poly-

definite integration The process of calculating (x) the nomial q. (3) For any expression of the form fg(x the definite integral of some function. function g(x). degenerate conic sections Special cases of conic sections, when the plane cutting the cone produces dense set A subset of real numbers is dense, if for a point or a pair of intersecting lines. Algebraically, any point a of that set there are infinitely many other these cases correspond respectively to the equations points b of the same set, arbitrarily close to a. Formally, the set S in R is dense, if for any point a ∈ S 2 2 and any number ε > 0, there exists another point y2 x y x2 + 2 = 0, y 2 = a, − 2 = 0. b ∈ S, such that |a − b| < ε. Examples: The set of all a2 b a2 b rational numbers is dense, but the set of all integers is not. degree of freedom If n unknowns (or variables) x1 , x2 , · · · , xn are connected by one relation, then density function Also called probability density only n − 1 of them can be chosen arbitrarily, whereas function. A function that represents some type of disthe nth one is dependent on the other choices. For tribution. The graph of a density function is called a that reason we say that the degree of freedom of these density curve. Density function must be non-negative variables is n−1. If the same unknowns are connected and that the area under a density curve is exactly 1, by two relations, then the degree of freedom would since it represents the total probability. Symbolically, a function f (x) is a density function if f (x) ≥ 0 and be n − 2, and so on. Z ∞ degree of a polynomial For a general polynomial f (x)dx = 1. given by the formula −∞

n

p(x) = an x + an−1 x

n−1

+ · · · + a1 x + a0 ,

dependent variable In an equation y = f (x) the variable y. The variable x is called the independent the whole number n ≥ 0. Example: For the polynovariable. mial p(x) = 3x5 − 2x4 + 5x2 − 6 the degree is 5. derivative Let the function f (x) be defined on degree measure of an angle One of the two main some interval (a, b). The derivative of the function units for measuring an angle (the other unit is called at some point x = c is defined as the limit radian measure). In this system the circumference of any circle is divided into 360 equal arcs and the f (x) − f (c) f 0 (c) = lim , angle subtended on one of those arcs is said to have x→c x−c

35 if this limit exists. The derivative of a function is a function itself given by f (x + h) − f (x) , h→0 h

f 0 (x) = lim

because we know that (sin x)0 √ −1 cos(sin x) = 1 − x2 .

=

cos x and

derivative of a vector function See differentiation of vector functions.

where x is any point in the domain of the function Descartes’ rule of signs Let the polynomial of such that this limit exists. The set of all these points degree n ≥ 1 be written in the standard form: is the domain of the derivative. Some other common p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 , notations for the derivative of y = f (x) are: and a0 6= 0. Then (1) The number of positive zeros of p is either equal to the number of sign changes of coefficients of p(x) or is less than that number In practice, derivatives are calculated by the differen- by an even integer; (2)The number of negative zeros tiation rules, rather than using the definition directly. of p is either equal to the number of sign changes Examples: of coefficients of p(−x) or is less than that number by an even integer. Example: The polynomial 1 x 0 x , (e ) = e . (xn )0 = nxn−1 , (tan−1 x)0 = p(x) = 2x3 − 5x2 + 6x − 4 has three sign changes, so 1 + x2 it may only have three or one positive zeros (roots). See also left-hand derivative, right-hand derivative. Next, p(−x) = −2x3 − 5x2 − 6x − 4, has no sign For derivatives of functions of several variables see changes, so p cannot have any negative zeros. partial derivatives, directional derivatives, normal determinant A number, associated with any derivatives. square matrix. For a given matrix A the notations derivative of a composite function See chain for corresponding determinant are |A| or det(A). The rule. value of the determinant is defined recursively as follows: If the matrix is 1 × 1, A = (a), then |A| = a. derivative of indefinite integral Let f be a conFor the 2 × 2 matrix tinuous function on some interval [a, b] and F (x) = Rx f (t)dt. Then F 0 (x) = f (x) for a < x < b. This rea b a , sult is one case of the Fundamental Theorem of Calc d culus. This statement has far reaching generalizations for functions f (x) other than continuous func- |A| = ad − cb. In general, for the n × n matrix tions. a11 a12 . . . a1n derivative of the inverse function Let f (x) be a21 a22 . . . a2n . a differentiable function on some interval I and let .. .. .. . . g(x) be its inverse. Then, if for some point a ∈ I, 0 an1 an2 . . . ann f (g(a)) 6= 0, then the derivative of g at that point is given by the determinant is equal to 1 g 0 (a) = 0 . f (g(a)) n X |A| = a1j C1,j , This formula makes it possible to calculate derivaj=1 tives of functions if we know the derivatives of their inverse. Example: where C1,j are the first row cofactors. This presentation is called expansion by the first row. It is possible 1 d (sin−1 x) = √ , to expand the determinant by any row or column and dx 1 − x2 y0 ,

dy df , , Dx f, Df (x). dx dx

36 the formulas are similar. Example: −2 2 3 1 −3 −4 = (−2) −3 −4 0 1 −4 0 1 1 −3 1 −4 + 3 −2 −4 0 −4 1 = (−2) · (−3) − 2 · 17 + 3 · 12 = 8. Equivalent definition of determinants is possible with the use of permutations and signed elementary products. With this approach, the determinant of the matrix is defined as the sum of all possible signed elementary products and the signs depend on the fact of the permutation being odd or even. determinant function The same as determinant of a square matrix. Equivalent definition: Determinant function is the sum of all signed elementary products.

that that eigenvalues be the elements on the diagonal of D. The method of diagonalization is also used in solving homogeneous or non-homogeneous differential equations. diameter For a circle, the line segment passing through the center an connecting two opposite points on the circumference. Standard formulas involving diameter are: d = 2r, r is the radius of the circle, C = πd, C is the length of circumference. difference When performing the operations of subtracting one number from the other, one function from the other, one matrix from the other, etc., the result is the difference of given objects. difference quotient For a given function f (x) the expressions f (x) − f (a) x−a

or

f (x + h) − f (x) , h

where x 6= a and h 6= 0. These expressions are the deviation In statistics, the difference between an important part of definition of the derivative of the individual value of a set and the mean of all set of function. values under consideration. See also standard deviadifference law of limits If the functions f (x) and tion and variance. g(x) both have finite limits when x approaches some diagonal In geometry, in case of polygons or poly- point a, then the limit of the difference function is hedra, diagonal is understood as any segment, con- also finite and necting the vertices of that objects. lim (f − g)(x) = lim f (x) − lim g(x). x→a x→a x→a diagonal matrix A matrix, that has its only nonzero elements on the main diagonal. The general di- See also limit laws. agonal matrix has the form difference rule For differentiation. If f (x) and a1 0 0 . . . 0 g(x) are differentiable, then 0 a2 0 . . . 0 [(f − g)(x)]0 = f 0 (x) − g 0 (x). 0 0 a3 . . . 0 . .. .. .. .. See also differentiation rules. . . . 0

0

0

...

an

differentiable function at a point x = c. A function, that is defined at a neighborhood of the point c diagonalization of matrices A square matrix A and for which the first derivative exists. If that propis diagonalizable, if there exists an invertible matrix erty is true for all points of some interval (a, b), then P such that P −1 AP = D and D is a diagonal matrix. the function is called differentiable on that interval. The main theorem in the problem of diagonalization differential (1) For functions of one variable. Let states that a matrix is diagonalizable if and only if it y = f (x) be a differentiable function. Then has a system of linearly independent eigenvalues. Additionally, the matrix P could be chosen in such a way df = dy = f 0 (x)dx

37 is called the differential. formulas. These are the rules that allow to calcu(2) For functions of several variables. Let f (x, y, z) late derivatives of functions easily, without relying be differentiable. Then on its definition. Here are the basic rules. Let f (x) and g(x) be two differentiable functions on ∂f ∂f ∂f df = dx + dy + dz some interval. Then for derivatives of their sum, dif∂x ∂y ∂z ference, product, and quotient we have: is the differential of f . Differential of a function of d d d two or n variables is defined similarly. In multidimen[f (x) ± g(x)] = f (x) ± g(x) dx dx dx sional case the differential is also called total derivative. d d d [f (x)g(x)] = g(x) f (x) + f (x) g(x) dx dx dx differential equation An equation that involves d d the unknown function along with its derivatives of g(x) dx f (x) − f (x) dx g(x) d f (x) = . some order. These equations are classified depending 2 dx g(x) [g(x)] on order of derivative, the type of coefficients and For derivative of composite functions see chain rule many other characteristics. The equation and for derivatives of implicit functions see implicit differentiation. 2x2 y 00 − sin xy 0 + ln y = 3x is a non-homogeneous second order linear ordinary differential equation with variable coefficients. In cases when the right side is zero (i.e., only the functions y, y 0 y 00 , etc. are involved), the equation is called homogeneous. If any of the ”variables” y, y 0 y 00 , · · · are present in a non-linear form, then the equation is called non-linear. When functions of several variables and their partial derivatives are involved, equation is called partial differential equation. The solution of differential equation is any function, that substituted into the equation makes it an identity. The general solution of an equation is a solution that contains all possible solutions.

differentiation of a vector function Let r(t) be a three dimensional vector function. The derivative of this function is defined very similar to the derivative of scalar functions: r0 (t) = lim

h→0

r(t + h) − r(t) h

if the limit exists. Vector differentiation follows the same rules as for scalar functions in the sense that addition, subtraction, scalar multiplication rules are identical. However, since there is no notion of division of vectors, we do not have quotient rule. Furthermore, multiplication has two versions for three dimensional vectors: scalar product and cross proddifferential operator An operation that assigns uct. The product rules in both cases are similar: If to a given function an expression containing that u(t) and v(t) are two vector functions, then function and some of its derivatives. The expression d L[φ](x) = φ00 + p(x)φ0 + q(x)φ, [u(t) · v(t)] = u0 (t) · v(t) + u(t) · v0 (t), dt where φ(x) is twice differentiable function is an examd [u(t) × v(t)] = u0 (t) × v(t) + u(t) × v0 (t). ple of second order differential operator. The order of dt the operator is determined by the highest derivative All these rules are also valid for vector functions in involved. any dimensional space except the last one because differentiation The process of calculating the the cross product is defined in R3 only. Additionally, derivative of some function. See also implicit differ- the chain rule has the following form: entiation, partial differentiation. d [u(f (t)) = f 0 (t)u0 (f (t)). differentiation rules Also called differentiation dt

38

digits In decimal numeric system the numerals 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. dilation operator Or dilation transformation. An operator that applied to a vector does not change its direction but makes its norm bigger. The operator is given by the formula T x = kx, where k ≥ 1. In the case 0 ≤ k ≤ 1 the transformation is called contraction operator. For a more general case when not only the norm but also the direction could be changed, see expansion operator.

function results in the value at point c: Z ∞ δ(t − c)f (t)dt = f (c). −∞

This function plays very important role in theoretical phisics.

direct variation The name of many possible relations between two variables. The most common are:(1) y varies directly with x means that there is a real number k 6= 0, such that y = kx; (2) y varies directly with x2 means y = kx2 ; (3) y varies directly dimension of vector space If a vector space with x3 means y = kx3 . There are many other posV has a system of linearly independent vectors sibilities but rarely used. Compare also with inverse {v1 , v2 , · · · , vn }, that span V , then the number n variation and joint variation. is the dimension of the vector space. Also, any basis directed line segment The same as vector. of V , has the same number of vectors. In the case, when there is no finite basis, the space is called infi- directional derivative For functions of two real variables. Let u = (a, b) be a unit vector in the plane nite dimensional. and f (x, y) be a function. The derivative of f in didimension theorem for linear transformations rection of the vector u at some point (x0 , y0 ) (denoted Similar to dimension theorem for matrices. If T is a by Du f (x0 , y0 )) is the limit linear transformation from the n-dimensional vector space V to some other vector space W , then f (x0 + ha, y0 + hb) − f (x0 , y0 ) , lim h→0 h rank(T ) + nullity(T ) = n. if it exists. The definition for functions of three or more variables is similar. dimension theorem for matrices If A is a ma- direction angles The angles any nonzero vector v trix with n columns, then makes with the coordinate axes are called direction angles. The cosines of these angles are the direction cosines. Hence, if α is the angle between v and x-axis, then Here rank(A) is the rank of the matrix and v·i cos α = nullity(A) is the dimension of the kernel of the ma|v||i| trix. and similarly for other two directions. If a vector is Dirac delta function A generalized ”function”, used to describe the direction of some line then the formally defined for any real point c by the condi- direction cosines are also called direction numbers. tions: direction field For differential equations such as Z ∞ the equation y 0 = f (t, y). We can make a direcδ(t − c) = 0, t 6= c, δ(t − c)dt = 1. tion field by choosing several points on the coordinate −∞ plane and by drawing a small line segment from each The Laplace transform of this function could be cal- point with the slope equal to the value of f at that culated in generalized sense and is equal to e−sc . point. This method is a useful tool when trying to Additionally, convolution of any function with Dirac get an idea about the behavior of the solution which rank(A) + nullity(A) = n.

39 √ the quantity D = b2 − 4ac is the discriminant. Depending on the sign of D, the corresponding directrix In geometric definition of the parabola, quadratic equation f (x) = 0 has: (1) Two distinct the name of the fixed line with the following propreal roots, if D > 0; (2) Two real repeated roots, if erty: A parabola is the set of all points on a plane D = 0 ,or (3) Two complex conjugate roots, if D < 0. that are equidistant from a fixed point (called focus) and a fixed line, not containing that point is called disjoint events Two events that cannot happen at directrix. In a more general view, directrix could be the same time. In symbolic form, if A and B are the defined also for ellipses and hyperbolas. See eccen- events, then they are disjoint if A ∩ B = ∅. tricity. disk The term is used as a synonym for circle. The discontinuity Property for a function, opposite to spelling disc is also valid. being continuous. A function could be discontinuous displacement Some kind of shift or move from the by different reasons. For example, the function given position. n 1 if x ≥ 0 f (x) = distance One of the most important geometric −1 if x < 0 (and also physical) notions. The distance is underis discontinuous at the point x = 0, because the stood as a measure of how far or close two objects left-hand and right-hand limits at that point are not are. Different measuring methods and also different equal. The function f (x) = 1/x is discontinuous at measuring units, such as metric or English measuring the point x = 0, because it is not defined there. See units, may be used. In Algebra and Calculus we meaalso discontinuous function. sure distances by the distance formula or by many discontinuous coefficients Usually applies to dif- other methods, such as arc length formula, involving ferential equations with variable coefficients. In many integration. See corresponding entries for details. cases these kind of equations can be solved (or shown distance between a point and a line If (a, b) to have a solution) even if the coefficients have some is some point on the plane and the equation Ax + type of discontinuity, usually jump discontinuity. By + C = 0 represents a line, then the distance of discontinuous function A function f (x) is dis- the point from the line is given by the formula continuous at some point x = c, if one of the following |Aa + Bb + C| . d= √ happens. (1) The limit limx→c f (x) does not exist. A2 + B 2 (2) The limit exists, but does not equal the value of the function at that point: limx→c f (x) 6= f (c). (3) The function is defined around that point, but not at distance between a point and a plane Let the equation of a plane be given by Ax+By +Cz +D = 0 the point. This property is opposite to the property of being and the point P (x0 , y0 , z0 ) is not on the plane. The distance now will be given by the formula continuous. Ax0 + By0 + Cz0 + D| discrete variable A variable that does not take √ d= . A2 + B 2 + C 2 its values continuously. Being discrete could be expressed in different ways. For example, if the variable x takes whole values 0, 1, 2, 3, · · · or it takes rational distance formula Let (x1 , y1 ) and (x2 , y2 ) be two values, then it will be discrete in both situations. As points on the plane. Then the distance between them a rule, however, we usually consider random variables is given by the formula p taking either whole or integer values. d = (x2 − x1 )2 + (y2 − y1 )2 . discriminant For the general quadratic function Similarly, if x = (x1 , x2 , · · · , xn ) and y = (y1 , y2 , · · · , yn ) are two vectors in Euclidean space Rn , f (x) = ax2 + bx + c is difficult (even impossible) to find.

40 then the distance is given by a similar formula v u n uX d = t (yk − xk )2 . k=1

distance problem The problem of finding the distance traveled by a moving object. If the object moves with a constant speed v, then the distance, s is given by s = v · t, where t is the time of travel. In general, if the velosity of the object is variable and is given by the function f (t), then the distance traveled between time periods t = a and t = b will be given by the integral Z s=

b

f (t) dt. a

distribution More precisely, probability distribution. For any random event, the outcomes appear with certain frequencies. The combination of all outcomes with their relative frequences is a distribution. Depending on types and number of outcomes, distributions are divided into discrete and continuous. Any discrete distribution has only finite number of outcomes and continuous distributions have infinitely many, but not discrete outcomes. (1) Discrete distributions. Every outcome has a probability (frequency) of appearing, that is either positive or zero. The sum of probabilities of all possible outcomes is 1. Formally, if there are n outcomes X1 , X2 , · · · , Xn and P (A) indicates the probability of an event A, then P (Xi ) ≥ 0, i = 1, 2, · · · , n and P P (Xi ) = 1. Binomial distribution is one of the most common discrete probability distributions and histograms are their most common visual representations. (2) Continuous distributions are represented by a density curve which is the graph of a positive function with the property that the area under that curve is 1. Individual probability of any outcome is always zero (because there are infinitely many outcomes), hence the probabilities in question are probabilities of groups of events. This means that if X is any random outcome, then P (X) = 0 but, for example

P (a ≤ X ≤ b) ≥ 0. Additionally, as in the case of discrete distributions, the probability of all space of outcomes is 1. For specific examples of continuous distributions see normal distribution, Poisson distribution, t-distribution, F-distribution, uniform distribution. distributive property For any real or complex numbers a, b, c, the property a(b + c) = ab + ac, that allows to group, or to split algebraic or numeric expressions. Similar properties are valid also for other mathematical objects, such as functions, matrices, etc. divergence of a vector field For a vector field F = P i + Qj + Rk in R3 the divergence is a scalar function, defined by the equation divF =

∂Q ∂R ∂P + + , ∂x ∂y ∂z

if the partial derivatives exist. See also curl of a vector field and gradient vector field. divergence test for series Let {an } be a sequence of numbers. If limn→∞ an P does not exist or ∞ if limn→∞ an 6= 0, then the series n=1 an is divergent. divergence theorem Let F = P i + Qj + Rk be a vector field with continuously differentiable components in some open region of R3 . Let D be a region,contained in the domain of F, with the boundary S. Then, if n is the outward normal to S, we have the relationship Z Z Z Z Z F · n dS = divF dV. S

D

Also called Gauss’ theorem. divergent R ∞ improper integral (1) The improper integral a f (x)dx is divergent, if the limit Z A lim f (x)dx A→∞

a

does not R b exist or is infinite. R ∞ The definitions for integrals −∞ f (x)dx and −∞ f (x)dx are similar.

41 (2) For improper integrals of unbounded functions Rb on bounded intervals a f (x)dx, where f has infinite limit at the point x = b. We call that integral divergent, if the limit Z b−δ lim f (x)dx δ→0

a

Theorem of Algebra, any polynomial of degree two or higher is divisible by another polynomial, with possibly complex coefficients. If we restrict division to be by real polynomials only, then not all the polynomials will be divisible. Examples: The polynomial p(x) = x2 − 5x + 6 is divisible by real polynomials x − 2 and x − 3. The polynomial q(x) = x2 + 4 is not divisible by any real polynomial but it is divisible by complex polynomials x + 2i and x − 2i.

does not exist or is infinite. The definitions for the cases when the point of unboundedness of the function is the left endpoint or some inside point, is sim- division One of the four basic operations in algebra and arithmetic. By dividing two numbers we get ilar. their ratio. Division is possible by any real (or comdivergent power series Each power series plex) number, except zero. Division of any number ∞ by zero is not defined. X cn (x − a)n division of complex numbers (1) To divide two n=0 complex numbers a+ib c+id , written in the standard form, has an interval of convergence given by |x − a| < R, we multiply both numerator and denominator by the where R ≥ 0. For any values of x, |x − a| > R, the conjugate of the denominator, c − id. This allows to series will be divergent. In case when R = 0, the se- get a real number c2 + d2 in the denominator and diries will be convergent in one point only, x = a. See vide the resulting complex number in the numerator by that real number: also divergent series and convergent series. divergent sequence A sequence, where the terms do not go to any particular finite limit. The opposite of the convergent sequence. Examples: The sequences 1, −1, 1, −1 · · · and 1, 4, 9, ..., n2 , ... are divergent. P∞ divergent series A numeric series n=1 an is divergent, if the sequence of its partial sums Sn = Pn i=1 ai does not have a finite limit, i.e. is a divergent sequence. The opposite of convergent series. Examples: The harmonic series is divergent, because its partial sums grow indefinitely large. The series 1 − 1 + 1 − 1 + 1 − · · · is divergent, because its partial sums do not approach any particular limit. The notion of divergence applies also to functional series such as power series or Fourier series. A functional series is divergent at some point if the numeric series formed by the values of functional series at that particular point is divergent.

a + ib ac − bd ad + bc = 2 + 2 i. c + id c + d2 c + d2 (2) If the complex numbers z = r(cos θ + i sin θ) and w = ρ(cos φ + sin φ) are given in trigonometric form, then z r = [cos(θ − φ) + i sin(θ − φ)]. w ρ division of fractions To divide the fraction a/b by another fraction c/d, we just multiply the first one by the reciprocal of the second: a c a d ad ÷ = · = . b d b c bc See also multiplication of fractions.

division of functions For two functions f (x) and dividend In the division process of two numbers, g(x) their quotient (division function) is defined to f (x) f the number, that is getting divided. The other num- be the quotient of their values: g (x) = g(x) . This function is defined where f and g are defined, except ber is called divisor. where g(x) = 0. divisible polynomial A polynomial, that can be divided by another polynomial. By the Fundamental division of polynomials Let P (x) and Q(x) be

42 two polynomials and assume that degree of Q is less or equal to the degree of P . To divide P by Q, means to find out ”how many times Q fits into P ”. As in the case of division of integers, the result is not always a specific polynomial, but rather there is some remainder. By the division algorithm, in this case we have R(x) P (x) = D(x) + , Q(x) Q(x) where R(x) is the remainder of this division. Example:

(x1 , x2 , · · · , xn ) and y = (y1 , y2 , · · · , yn ) in Euclidean space Rn , the number x · y = x1 y1 + x2 y2 + · · · + xn yn . Dot product is a special case of inner product. See also scalar product. double angle formulas For trigonometric functions the formulas sin 2x = 2 sin x cos x cos 2x = cos2 x − sin2 x =

2x + 2 5x3 − 4x2 + 7x − 2 = 5x − 4 + 2 . x2 + 1 x +1

2 cos2 x − 1 = 1 − 2 sin2 x 2 tan x . tan 2x = In practice, division process is done either by long 1 − tan2 x division , or by synthetic division algorithms. The second one is possible only if we need to divide by a Similar formulas are valid for other trigonometric binomial of the form x − c. See corresponding entries functions but are rarely used. for the details of the algorithms. double integral (1) Let a function f (x, y) of two variables be defined on some rectangular region D = divisor In the division process of two numbers, the [a, b] × [c, d] = {(x, y)|a ≤ x ≤ b, c ≤ y ≤ d}. Then number, that divides the other one, called dividend. the double integral of the function in the domain D domain of a function For a function of one real is the limit of its Riemann sums, as the number of variable f (x), the set of all real values x, such that the partitions of the sides of rectangle goes to infinity: function is defined and is finite. More generally, for a Z Z Z Z function f (x1 , x2 , · · · , xn ) of n independent variables, f (x, y)dA = f (x, y)dxdy the region G in the Euclidean space Rn , such that f D D is defined for all values of variables that belong to the n X m X region G. = lim f (x∗i , yj∗ )∆x∆y. n,m→∞ Examples: For the function of one variable i=1 j=1 f (x) = √

1 x−1

(2) In the case the function is given on a more general region than a rectangle, the definition requires some modifications. Assume that the region has the form

the domain is the set {x|x > 1} because the square root function is defined for non-negative values of argument and, additionally, division by zero is not defined. For the function of two variables f (x, y) = ln(x2 + y 2 )

D = {(x, y)|a ≤ x ≤ b, g1 (x) ≤ y ≤ g2 (x)}, where g1 , g2 are continuous on [a, b]. Then the double integral on this type of regions is defined to be the iterated integral Z Z Z b Z g2 (x) f (x, y)dA = f (x, y)dydx.

the domain is all values of x and y except (0, 0) D a g1 (x) because the logarithmic function is not defines at (3) Integrals over the regions of the type the origin. dot

product

For

any

two

vectors

x

=

D = {(x, y)|c ≤ y ≤ d, h1 (y) ≤ x ≤ h2 (y)}

43 are defined similarly. See also definite integral, triple integral, multiple integral. double Riemann sum Let a function f (x, y) of two variables be defined on some rectangular region D = [a, b] × [c, d] = {(x, y)|a ≤ x ≤ b, c ≤ y ≤ d}. In analogy of forming Riemann sums, we divide interval [a, b] into n equal parts of size ∆x = (b − a)/n and interval [c, d] into m equal parts of size ∆y = (d − c)/m by choosing points x0 (= a), x1 , · · · , xn (= b) and y0 (= c), y1 , · · · , ym (= d). In resulting smaller rectangles Di,j = [xi−1 , xi ] × [yj−1 , yj ], 1 ≤ i ≤ n, 1 ≤ j ≤ m, we choose some points (x∗i , yj∗ ) and form the double sum m n X X f (x∗i , yj∗ )∆x∆y, i=1 j=1

which is called double Riemann sum. The choice of points (x∗i , yj∗ ) is arbitrary, and (under some restrictions on function) always results in the same limit. See also double integral.

E e One of the most important universal constants in mathematics. This is a non-algebraic, transcendental number, such that there are many approximation formulas. Among them, most familiar are: e=

∞ X 1 n! n=0

and

1 n ) . n The decimal approximation of e is 2.7182818... e = lim (1 + n→∞

eccentricity For conic sections. Let L be a fixed line on the plane (called directrix ) and P be some point (called focus), outside of that line. A positive real number e is now fixed. The set of all points Q on the plane, satisfying the relation dist(Q, P ) =e dist(Q, L) is called conic section and e is its eccentricity. In case e < 1 the result is ellipse, for e = 1 we get parabola and when e > 1, the result is hyperbola. For ellipses and hyperbolas the eccentricity has a simpler description. See respective definitions. echelon form of a matrix Also called row echelon form. The result of performing Gaussian elimination process on the matrix. In echelon form each row should start with zero or the leading one. Further, in each next row the leading one should be to the right of the previous row’s one. Finally, the rows with all zero elements should appear in the bottom rows only. Example: The matrix 1 0 0 0

4 0 0 0

0 1 0 0

−3 0 1 0

is in row echelon form. The matrix is said to be in reduced row echelon form, if, additionally, the only

44 non-zero entries are the leading ones.

elementary product For a square matrix A of size n, elementary product is any product of n elements eigenspace Let λ be an eigenvalue of a given from that matrix that contains exactly one element square matrix A and assume {x1 , · · · , xk } are all linfrom each row and each column. See also signed eleearly independent eigenvectors of that matrix cormentary product. responding to the same eigenvalue. Linear space spanned by that vectors is called the eigenspace for elementary row operations For systems of algethe eigenvalue λ. braic linear equations or matrices the three elementary row operations are: eigenvalues of matrix For a square matrix A, a (1) Interchanging any two rows; real or complex number λ, such that there exists a (2) Multiplying any row by any non-zero constant; non-zero vector x satisfying the equation Ax = λx. (3) Adding non-zero multiple of any row to any other Eigenvalues are roots of the characteristic equation. row. An eigenvalue is called simple if it appears only once These operations are performed as a part of eliminain the factorization of the characteristic polynomial. tion method of solving systems of linear equations. There are two different notions of multiplicity for eigenvalues. The number of times λ − λ0 appears in elimination method One of the methods of solvthe factorization of the characteristic polynomial is ing systems of equations with more than one variits algebraic multiplicity. On the other hand, the di- able. The method constitutes of eliminating one of mension of the eigenspace, corresponding to the value the variables in one or more equations, thus arrivλ0 is its geometric multiplicity. According to a theo- ing to equation (or system of equations) with less rem, the geometric multiplicity is always less than or variables. This approach is mainly used for solving equal to its algebraic multiplicity. See also eigenvec- systems of linear equations but can also be used for tors of matrix below. solving non-linear systems. For specific description of this method for solving systems of linear equations, eigenvectors of matrix A non-zero vector x, satsee Gaussian elimination or Gauss-Jordan eliminaisfying the equation Ax = λx for some square matrix tion. A and real or complex number λ. The number λ is called an eigenvalue. Eigenvectors, corresponding to ellipse One of the three main conic sections. Gedifferent eigenvalues are orthogonal (and hence, lin- ometrically, an ellipse is the location (locus) of all early independent). See also eigenvalues of matrix. points in a plane, the sum of whose distances from two fixed points in the plane is a positive constant. element of a matrix The numbers (or functions) The two points, from which we measure distances, are that form a matrix. See also entries of a matrix. called foci (plural for focus). Equivalently, the ellipse element of a set Members of the given set. De- could be described as the result of cutting double cone pending on nature of the set the elements can be num- by a plane not parallel to its axis or any of the genbers, functions, matrices or any other mathematical erators of the cone, and not passing through vertex objects. of the cone. In the special case, when the plane is perpendicular to the axis, the result is a circle. An elementary matrices Any matrix, which could be ellipse is a bounded smooth curve. Alternative georeceived from the identity matrix by one elementary metric definition could be given with the use of ecrow operation. Example: The matrix centricity. See corresponding definition. 1 0 0 Algebraically, the general equation of an ellipse is 0 1 0 given by the quadratic equation Ax2 + Bxy + Cy 2 + 1 0 1 Dx + Ey + F = 0, where A, B, C, D, E, F are real constants and A · C > 0. In the case, when the is elementary, because it is the result of adding the foci are located on one of the coordinate axes and first row to the third row of the identity matrix.

45 center coincides with the origin, this equation could be transformed into standard form y2 x2 + = 1, a2 b2 where the constants a, b have special meaning: c = √ a2 − b2 is the distance from origin to a focus.

ellipsoid The three dimensional surface given by the formula x2 y2 z2 + 2 + 2 = 1. 2 a b c elliptic cone The three dimensional surface given by the formula

Moreover, the intersection points of ellipse with the coordinate axes are called vertices of ellipse and their distances from origin are equal to a and b. The line connecting two foci is called major axis and the perpendicular line (passing trough origin) is the minor axis. In the more general case, when the center of ellipse is located at some point (h, k) but the major and minor axes are still parallel to coordinate axes, the standard equation becomes 2

2

(x − h) (y − k) + = 1. a2 b2 In these cases eccentricity is given by the formula e = c/a. Both of the above cases happen when in the general equation the term Bxy is missing. In the case B 6= 0 the result is still an ellipse, which is the result of rotation of one of the previous simpler cases. The ellipse could also be given by its polar equation: r=

de 1 ± e cos θ

or r =

de , 1 ± e sin θ

where d > 0 and 0 < e < 1 is the eccentricity of the ellipse.

x2 y2 z2 + − = 0. a2 b2 c2 This surface represents a cone which perpendicular cuts are ellipses instead of circles. In particular case a = b we get circular cone. elliptic paraboloid The three dimensional surface given by the formula x2 y2 z + = . a2 b2 c The picture shows the paraboloid given by the equation z = x2 + y 2 .

46 empty set The set, that contains no elements. The equation A statement that one quantity equals annotation is ∅. other. Unlike identities, equations are not true for all the values of variables that are involved in that endpoints For the closed or open intervals relationship and by that reason they are also called [a, b], (a, b), the points a and b. The same is true conditional equality. To solve an equation means to for half-open intervals [a, b) and (a, b]. find all values of the variables or unknowns that make endpoint extreme value Minimum or maximum that statement an identity. Depending on the type value of a function, that occurs at the endpoint of the of variables, the equations can be polynomial, ratioclosed interval [a, b]. nal, trigonometric, radical, exponential, logarithmic, and so on. In the case when the unknown quantity English system of measurements The system is a function, then we can have a differential, integral where the distances (also areas and volumes) are meaor integro-differential equation. There are also equasured by inches(in), feet(ft), yards(y), and miles(mi) tions that contain matrices, vectors and other mathand the weights are measured using ounces(oz) and ematical objects. The equations could be given both pounds(lb). Unlike the metric system of measurein rectangular coordinates and also in polar, spherical ments which is based on decimal numeric system, reand other variables. Examples: lations in English system are more or less arbitrary. Polynomial: 3x3 − 2x2 + 4x − 5 = 0; Here are some of that relations: 1 ft= 12in; 1 y=3 ft, Exponential: 2x−1 − 4 = 0; 1 mi= 1760 y , 1 lb= 16 oz. Trigonometric: sin x + cos x = 1; entries of matrix Same as elements of a matrix. Logarithmic: ln(x2 − 1) + ln(x2 + 1) = 0; Polar: r2 + sin θ = 0; epicycloid When a circle rolls on the outside of a Differential: 2xy 00 + 3x2 y 0 − y = 0. fixed circle, the path of any point is called epicycloid. Some equations cannot be categorized because they Let a be the radius of the fixed circle with center at contain different kinds of functions, such as the equathe origin and b be the radius of the moving circle. tion 2x2 − 2x+1 + sin x = 0. These kind of equations When we connect the origin (center of the first circle) are called mixed. See also corresponding entries for with the center of the second circle and continue, the all the mentioned types of equations. resulting radius in the moving circle will form some angle θ with the horizontal line. In this setting, the equilateral triangle A triangle that has all sides parametric equations of epicycloid are given by the equal in size. In equilateral triangles all three angles formulas are also equal in size and measure 60◦ (π/3 in radian measure). a+b θ, x = (a + b) cos θ − b cos equilibrium point or position In Physics or b Mechanics the point or position where the system a+b reaches equilibrum. This means that after reaching y = (a + b) sin θ − b sin θ. b that point the system does no longer move unless equality The property of being equal to. Math- some outside force is applied. ematically, equality means that certain quantity is equilibrum solution For autonomous differential equal another. These quantities may contain numequations. When solving the equation y 0 = f (y), bers, variables, functions, and other mathematical some solutions turn out to be constants, hence, do objects. If the equality holds for all values of given not depend on time, and are called equilibrum. This parameters, then we have the case of identity. If the happens exactly at the roots of equation f (y) = 0 equality is true for some values of that parameters and these roots are called critical points. only, then we have the case of equation. Equations are also called conditional equality. See correspond- equivalence A notion that has many manifestaing entries. tions in mathematics. In the most general and ab-

47 stract form it could be defined as some kind of relation between elements of a set S that satisfies three axioms for any elements a, b, c of that set. If we denote that relation by ”◦”, then the axioms are: 1. (Reflexivity) a ◦ a 2. (Symmetry) if a ◦ b then b ◦ a 3. (Transitivity) if a ◦ b and b ◦ c then a ◦ c . This way equality relation becomes equivalence, similarity of triangles is equivalence as well as others.

gent. Denote by S its sum and by SN its N th partial sum. Then |S − SN | < |aN +1 |. See also alternating series test and alternating series estimation theorem.

error Any time an exact value (of a number, function, series, integral, etc.) is substituted by an approximate value, an error occurs. Errors could be estimated in absolute terms (such as the difference between exact and approximate values) or in relative terms (as a proportion of error to the exact value). For different specific error estimates see the entries following this one on error estimates of series and integrals.

This estimate is very similar to the estimate for the Midpoint rule.

error estimate for the Midpoint rule Assume that we wish to calculate numerically the integral Rb f (x)dx using the Midpoint rule and denote by a Mn the nth approximation using that rule. Then, if |f 00 (x)| ≤ M for all a ≤ x ≤ b, then Z M (b − a)3 b equivalent equations Two equations that have . f (x)dx − T 2 n ≤ the same solution sets. Equations x −5x−6 = 0 and a 24n2 (x + 1)(x − 6) = 0 are equivalent, because they have the same solutions x = −1, 6. The term is usually This estimate is very similar to the estimate for the used for equations of the same type, but equations of Trapezoidal rule. different nature can be equivalent too. The equation error estimate for Simpson’s rule Assume 2x − 1 = 1 is algebraic and log2 x = 0 is logarithmic, that we wish to calculate numerically the integral and they are equivalent because they have the same R b f (x)dx using the Simpson’s rule and denote by a solution x = 1. Sn the nth approximation using that rule. Then, if equivalent inequalities Two inequalities that |f (4) (x)| ≤ M for all a ≤ x ≤ b, then have the same solution sets. Inequalities x2 −5x−6 > Z b M (b − a)5 0 and (x + 1)(x − 6) > 0 are equivalent, because they . f (x)dx − S ≤ n have the same solution set. a 180n4 equivalent matrices Or row-equivalent matrices. If a matrix could be transformed to another matrix error estimate for the Trapezoidal rule Asby a finite number of elementary row operations, then sume that we wish to calculate numerically the inteRb the matrices are called equivalent. gral a f (x)dx using the Trapezoidal rule and denote equivalent system Two systems of equations that by Tn the nth approximation using that rule. Then, have the same solution sets. See also equivalent equa- if |f 00 (x)| ≤ M for all a ≤ x ≤ b, then tions Z b M (b − a)3 equivalent vectors Two vectors that have the f (x)dx − Tn ≤ . a 12n2 same direction and length.

error estimate for alternating series Assume P∞ that the series n=1 an is alternating and is conver-

estimate of the sum of a series The sums of most of the series are difficult or even impossible to calculate. Estimates serve as a substitute for exact value and are good enough if the error is small. See also error estimate for alternating series. Euclidean distance Distance, generated by the Euclidean norm. For two vectors x = (x1 , x2 , · · · , xn ), y = (y1 , y2 , · · · , yn ) in Euclidean space, their distance is defined to be the

48 θ = π, we get the famous formula eiπ = −1 which connects all the fundamental numbers of mathematEuclidean inner product For two vectors in ics. the Euclidean space x = (x1 , x2 , · · · , xn ), and y = (y1 , y2 , · · · , yn ), the quantity Euler method For approximation of solutions of initial value problems. The idea behind this method x · y = x1 y1 + x2 y2 + · · · + xn yn . is to substitute the solutions of the problem (which usually are difficult or even impossible to find) by a It is the same as dot product. See also scalar product. linear function, which is the tangent line to this soEuclidean norm For a vector x = (x1 , x2 , · · · , xn ) lution at some point. By this reason the method is in Euclidean space the number ||x|| = (x21 +x22 +· · ·+ also called tangent line method. In detail, assume we x2n )1/2 . This number is non-negative and it is zero if are solving the problem and only if the vector is the zero vector. Also called y 0 = f (t, y) y(t0 ) = y0 magnitude or length of the vector. norm of their difference: dist(x, y) = ||x − y||.

Euclidean space The vector space of all vectors in some interval. As a first approximation we tale the with n coordinates, where the addition and scalar linear function multiplication operations are defined in the usual y = y0 + f (t0 , y0 )(t − t0 ) way: which passes through the point (t0 , y0 ) and hence, u + v = (u1 + v1 , u2 + v2 , · · · , un + vn ), coincides with the solution at that particular point. Next, we choose a point t1 not too far from t0 to αu = (αu1 , αun , · · · , αun ) assure that the error is small and construct another for any two vectors u, v and any real number α. Ad- line ditionally, the inner product is defined in this space y = y1 + f (t1 , y1 )(t − t1 ), as the dot product (or, which is the same, Euclidean where y1 = y0 + f (t0 , y0 )(t1 − t0 ), is the value found inner product). This, on its turn, induces the norm from the first line. Proceeding this way, we will have in the space, which defines Euclidean distance. This a series of equations special vector space is now called Euclidean space and is commonly denoted by Rn , where n is any positive y = yn + f (tn , yn )(t − tn ), integer. each of which is given on the interval [tn , tn+1 ]. These Euler’s constant The limit approximate solutions converge to the exact solution 1 1 1 when the lengths of intervals approach zero and unγ = lim (1 + + + · · · + − ln n), der certain conditions on function f . n→∞ 2 3 n which exists and is finite. The value of γ is approxi- Euler-Fourier formulas Also called Fourier formately 0.577212 but it is not known if the number is mulas. These formulas represent Fourier coefficients of the function by certain integral formulas involving rational or not. See also harmonic series. the function itself. Let f (x) be defined on some inEuler equation See Cauchy-Euler equation. terval [−L, L]. Then its Fourier coefficients are given by Euler’s formula The formula Z nπx 1 L eiθ = cos θ + i sin θ, f (x) sin dx, n = 0, 1, 2... an = L L √ −L where i = −1 is the imaginary unit. This formula Z is true also when the real number θ is substituted by 1 L nπx b = f (x) cos dx, n = 1, 2, 3... n any complex number z. In the particular case when L −L L

49 F (x, y) on R, such that evaluation The term is used in mathematics in different situations, most commonly when we want to find the value of a function. In this case, to evaluate a function means to find its value for certain value of the independent variable(s). Example: To evaluate the function f (x) = 3x4 − 2x2 + x − 4 at the point x = 2 means to calculate the value of f at that point: f (2) = 3 · 24 − 2 · 22 + 2 − 4 = 38. even function A function f (x) of real variable, that satisfies the condition f (−x) = f (x). This condition means that the graph of the function is symmetric with respect to the y-axes. The functions f (x) = cos x and f (x) = x2 are examples of even functions.

∂F = M, ∂x

∂F = N. ∂y

(1)

Every solution of this equation is given by the implicit function F (x, y) = c, c = constant. The necessary and sufficient condition for the equation being exact is My (x, y) = Nx (x, y). (2) Example: To solve the equation (3x2 − 2xy + 2)dx + (6y 2 − x2 + 3)dy = 0 we first easily check that the above conditions are satisfied. Next, to find the function F we integrate the first of equations (1) with respect to the variable x and get Z F (x, y) = (3x2 − 2xy + 2)dx = x3 − x2 y + 2x + g(y), where g(y) is an arbitrary function of the variable y. To satisfy the second of the equations (1) we differentiate this function by y and equate it to the function N: −x2 + g 0 (y) = 6y 2 − x2 + 3,

from where, g 0 (y) = 6y 2 + 3, or g(y) = 2y 3 + 3y (no need of arbitrary constant). Finally, solution of the even permutations A permutation that is the re- equation will be given by the implicit equation sult of even number of transpositions. F (x, y) = x3 − x2 y + 2x + 2y 3 + 3y = C. event A term used in Probability theory and Statistics to indicate some kind of outcome for the existence and uniqueness theorems For differ(usually random) variable. Tossing a coin and getting ential equations. Statements, that contain conditions Tail or Head are two different events. If an event can- under which the given differential equation has a solunot be split into any smaller events, then it is called tion and that solution is unique. Depending on type simple event. The collection of all simple events forms of equation (order, coefficients, homogeneous or not, the event space. Events can be dependent, indepen- boundary or initial conditions, etc) the existence and dent, mutually exclusive. See corresponding entries uniqueness conditions differ from each other. For example, if the equation is of the first order, then one for more information. boundary or initial condition is usually enough, while exact equations Let M (x, y) and N (x, y) be func- for the second order equations at least two conditions tions on some rectangle R = [a, b] × [c, d]. The first are necessary. Similar observations are valid for sysorder ordinary differential equation tems of equations too. Example: 0 Theorem. Assume we have the equation M (x, y) + N (x, y)y = 0 is called exact, if there exists a differentiable function

y 00 + p(t)y 0 + q(t)y = g(t)

50 with initial conditions y(t0 ) = y0 , y 0 (t0 ) = y00 where pressed with the use of exponential function with negall the functions p, q and g are continuous on some ative exponent: interval I, containing point t0 . Then there exists exf (t) = Aekt , k < 0. actly one solution y = φ(t) of this problem and the solution exists on all the interval I. Many economic, biological and other phenomena expanded form A term used in various situations. have exponential decay model. For a polynomial p(x) = (x − 2)(x + 5) the expanded form will be p(x) = x2 + 3x − 10. The term is used exponential function The function f (x) = ax , also for determinants, when expanded form means where a > 0, a 6= 1, which is defined for all real the form when it is expanded as a sum of cofactors values of x. The range of the function is (0, ∞). or minors. expansion by minors One of the methods of calculating the value of a determinant. See corresponding entry. expansion operator A linear operator that ”expands” given vector. In the most general setting, an expansion operator with factor k > 0 is a operator that makes the norm of a vector x be equal k||x||. Example: Operator given by a matrix k 0 exponential growth A functional model that is 0 k expressed with the use of exponential function with transforms any vector (x, y) into vector k(x, y). See positive exponent: also contraction operator, dilation operator. f (t) = Aekt , k > 0. expected value For a distribution, given by the Many economic, biological and other phenomena function f (x), the quantity have exponential growth. Z ∞ exponential order Functions that have growth or µ= xf (x)dx, decay comparable with the growth or decay of the ex−∞ ponential function. The sinh x and cosh x functions if it is finite. Expected value is the same as the mean both have exponential order. See hyperbolic funcof the distribution. This quantity could be defined tions. also for distributions of several variables. exponentiation One of the main mathematical experimental study In statistics, a study where operations. The exponent of a number b, b > 0 the researcher controls, modifies, or changes the con- (called base) of some power x is denoted by bx . In ditions in order to collect data of the results and ef- the case when x is a natural number, this expression fects of these changes. Most of the medical studies is understood as the number b multiplied by itself x are experiments to assess the results of treatment or times: bx = b · b · · · b. For exponents other than natural numbers, this definition is extended in several medication. steps. explanatory variable In statistics, another name (1) By definition b0 = 1 for any b > 0. for the independent variable. (2) If n is a positive integer, then b−n = 1/bn by defexponential decay A functional model that is ex- inition.

51 (3) For positive integers n, we define the exponent b1/n to be the nth root of the number b, i.e. that is a number that raised √ to the nth power (which is natural) results in b: ( n b)n = b. (4) For any integers n and √ m the exponent bn/m is √ m m n n defined to be the number ( b) = b . (5) To define the exponent bx for irrational number x we use limiting process: This power is defined as the limit bx = lim bcn , n→∞

of the data based on a sample data. extreme value Maximum or minimum values of a function. extreme value theorem If f (x) is a continuous function on the closed interval [a, b], then there exist points c, d ∈ [a, b] such that f (c) is the minimum and f (d) is the maximum values of f on [a, b]. A very similar statement is true also for functions of two or more variables.

extremum where {cn } is some sequence of rational numbers that maximum. approaches x. In definition of exponents we usually avoid the case b = 1 because it results in trivial case. From this definition the following important properties of exponentiation follow: (a) an · am = an+m (b) (an )m = an·m (c) (ab)n = an · bn (d) an /am = an−m . See also exponential function exponents at the singularity In differential equations. When solving an equation by the power series method and the point is regular singular point, the roots of the corresponding indicial equation are called exponents at the singularity. expression In mathematics, any combination of mathematical objects (numbers, functions, matrices, transformations, etc.) and operations on them. See also algebraic expression and arithmetic expression. extraneous root or solution In solving certain types of equations, some roots, that turn out invalid. This happens in solving rational, radical, logarithmic, and some other types of equations. Example: The equation √ 2x + 7 − x = 2 has two solutions, x = 1, x = −3 and the second one is extraneous. extrapolation A procedure of finding values of functions or data based on some known values of the function or given data. Similar to the notion of interpolation. The least square regression line is an example of extrapolation when we predict the values

Collective name for minimum and

52

F

collection) of functions is present. This fact is usually indicated by the presence of one or more parameters. By changing these parameter we get new functions, which, as a rule, do not change the nature of func2 F-distribution The distribution of the ratio of tions. Examples: (1) The family of functions Ax +B variances of two normally distributed populations. depends on two parameters and by changing the valThis distribution depends on both degrees of freedom ues of A and B we get different functions but all of of that normal distributions, hence there is not one them are quadratic. (2) When integrating the function cos x the result is sin x + C, which is a family but a family of F-distributions. of functions depending on one parameter. Change of factor If a number or function a is written as a C creates a new function but it is still a sinusoidal product a = b·c, then the numbers or functions b and function. c are called factors. A number or function may have more than two factors. The most common functions family of solutions When solving differential equations the solution usually depends on one or we factor are polynomials. See also factoring. more parameters creating a family of solutions. To factor tree Assume we need to factor a natural have a particular solution to a given equation it is number into the product of prime factors. Then, by necessary to impose certain restrictions on function, the fundamental theorem of arithmetic, the number called initial or boundary conditions. See also initial n could be written as a product pk11 · pk22 · · · pkmm . value problems, boundary value problems. Another way of writing this product is to let it expand vertically, and the resulting presentation is feasible solution Any solution of a linear procalled factor tree. Example: 84 = 22 · 3 · 7 could be gramming problem (a system of linear inequalities) is called feasible solution. The set of all feasible sowritten also as a factor tree lutions is the feasible region. As a rule, the feasible region is some kind of (possibly infinite) polygon. Fermat’s theorem Let the function f (x) be defined on the interval [a, b]. Assume that in some point c inside that interval the function gets its maximum or minimum value. If the function is differentiable at that point, then f 0 (c) = 0.

factorial of a whole number n ≥ 1 is defined to be the product 1·2·3 · · · n, denoted by n!. By definition, 0! = 1. factoring (1) The procedure of representing a number as a product of factors. Example: 24 = 3 · 8. See also prime factorization. (2) The procedure of representing a polynomial as a product of simpler polynomials. Example: 2x2 +5x−12 = (2x−3)(x+4).

Fermat’s Last Theorem If n > 2 is an integer, there are no positive integers x, y, z such that xn + y n = z n . This statement more properly should have been called conjecture, because Fermat did not prove it. It was proven more than 300 years later, in mid 1990s by A.Wiles. Fibonacci sequence (numbers) The sequence of numbers defined by the following rule: First two terms of the sequence are ones. Every other term is the sum of the previous two terms. First few Fibonacci numbers are: 1, 1, 2, 3, 5, 8, 13, 21, 34, . . ..

family of functions The term is used in cases when instead of one particular function a group (or field (1) A set of numbers where two algebraic oper-

53 ations of addition and multiplication are defined and satisfy standard rules (axioms) of that operations. In particular, both operations are commutative and associative and together they follow distributive rule. The most common numeric fields are the fields of rational numbers, real numbers and complex numbers. The set of integers (or positive integers) is not a field, because reciprocals are not integers. (2) More precisely, vector field. This notion is defined for vectors on the plane, space, or even higher dimensional spaces. For exact definition see vector field and for specific vector fields see also conservative vector field, gradient vector field, curl of a vector field.

fixed point of a function Any point x = a in the domain of some function f that satisfies the equation f (a) = a. If the function is viewed as a mapping, then the fixed points are the ones that do not move during the mapping f . Not all functions have fixed points and other functions may have more than one fixed point. Examples: The function f (x) = ex does not have any fixed points because the equation ex = x has no solutions. The function f (x) = x2 + x − 4 has two fixed points x = ±2, which are the solutions of the equation x2 + x − 4 = x. flux Let F be a continuous vector field defined on an oriented surface S with unit normal n. Then the surface integral of F over the surface S

finite dimensional vector space A vector space that has only finitely many linearly independent vectors. See dimension of vector space. Z Z first derivative test Let f (x) be defined on some interval [a, b] and in some inner point c, a < c < b, f 0 (c) = 0. (1) If f 0 (x) > 0 for c − δ < x < c and f 0 (x) < 0 for c < x < c + δ, then f has local maximum at point c. (2) If f 0 (x) < 0 for c − δ < x < c and f 0 (x) > 0 for c < x < c + δ, then f has local minimum at point c. (3) If f (x) does not change the sign while passing through point c, then the test cannot give definitive result.

F · n dS S

is called the flux of F across S.

focus of a conic section All three major conic sections have one (parabola) or two (ellipse and hyperbola) points that are important in geometric defifirst order differential equation Any differential nitions of that conic sections. For the definitions and equation that contains only the first derivative of the details see ellipse, hyperbola, parabola and also eccenunknown function. Depending on the other compo- tricity. Plural for the word focus is foci. nents of the equation is classified as linear, nonlinear, foil method of multiplication of two binomials. To with constant or variable coefficients, and also some multiply (a + b)(c + d), we multiply (F) first terms others. A typical first order linear equation with vari- a · c, (O) outside terms a · d, (I) inside terms b · c, (L) able coefficients is last terms b · d, and add them up: y 0 + p(t)y = g(t), because the function y is present in the first degree only and the coefficients p, q are functions, not constants. For specific equations of first degree see exact equations,Bernoulli equation, separable equations. five number summary In statistics. The five numbers for some data set are the minimum value, the maximum value, the median, and the first and third quartiles. These five numbers are important for construction of the boxplot.

(a + b)(c + d) = ac + ad + bc + bd.

folium of Descartes A plane curve given by the cubic equation x3 + y 3 = 3axy, where a is some real parameter. This curve has a slant asymptote given by the equation x + y + a = 0.

54 equation r = a sin 2θ, a > 0. The curve is called so because it consists of four loops. It is a special case of more general family of polar curves given by the equations r = a sin nθ, r = a cos nθ, where n is an arbitrary integer. The graph shows the equation r = sin 2θ.

force One of the most important notions in Physics. By the Newton’s second Law, if an object of mass m moves along a line with the position function s(t), then the force on the object is the product of the mass and its acceleration: F =m

d2 s . dt2

forced vibrations When the vibration of a spring or a pendulum is affected by some additional force F (t), then this vibration obeys the following differential equation: m

dx d2 x +c + kx = F (t). dt2 dt

formula An equation that contains two or more variables. As a rule, a formula is deduced to calculate values of one variable if the values of other variable(s) are known. Elementary formulas in most cases are interchangeable in a sense that they could be solved for any other variable. This is not true for more complicated formulas. Here are a few familiar formulas from geometry and algebra. See corresponding entries for more details. Area of a rectangle: A = ` × w. Compound interest: P = A(1 + r/n)nt . Distance formula: p d = (x1 − x2 )2 + (y1 − y2 )2 .

Fourier coefficients (1) In the Fourier series a0 /2 +

∞ X

(an cos nθ + bn sin nθ)

n=1

the coefficients an , n ≥ 0 and bn , n ≥ 1. (2) In a more general setting, if the function f (x) is defined on interval [0, 1], r(x) is some weight function, and φ1 (x), φ2 (x), · · · , φn (x) is an orthonormal set, then the numbers Z ak =

1

f (x)φi (x)r(x)dx,

i = 1, 2, ..., n,

0

are the Fourier coefficients of f . Fourier series For the given integrable function f (x) on the interval [0, 2π], the formal series f (x) = a0 /2 +

∞ X

(an cos nθ + bn sin nθ),

n=1

where the coefficients are defined by the formulas Z 1 π a = f (x) cos nxdx, n = 0, 1, 2, ... n four-leafed rose The plane curve, given by polar π −π

55 bn =

1 π

Z

π

f (x) sin nxdx, n = 1, 2, 3... −π

fraction Any real number that could be written n in the form m , where n and m are arbitrary real numbers with only one condition that m 6= 0. In the particular case when n and m are integers fractions are called rational numbers. For operations with fractions see addition and subtraction of fractions, multiplication of fractions, division of fractions. See also complex fractions and partial fractions for special types of fractions. frequency For trigonometric functions f (x) = a sin ωt and g(x) = a cos ωt, the number ω/2π is the frequency. This number shows how many times the functions complete one full cycle when we move the distance equal to 2π. The reciprocal to the frequency is the period of the function. Fresnel functions

The functions Z x S(x) = sin t2 dt 0

and

x

Z

cos t2 dt.

C(x) = 0

The integrals on the interval [0, ∞) are both converp gent and are equal to π/8. Frobenius method The method of solving linear differential equations near a regular singular point. See the entry series solution for the details. Fubini’s theorem In its most general form this theorem gives conditions on function of several variables which allow to calculate multiple integrals as iterated integrals. In the simplest form the theorem for double integrals is formulated as follows: Let f be a continuous function on the rectangle R = {(x, y)|a ≤ x ≤ b, c ≤ y ≤ d}. Then Z

b

"Z

f (x, y)dA = R

d

# f (x, y)dx dy

a

c

d

"Z

b

# f (x, y)dy dx.

= c

Fourier series of a function converges to that function if it satisfies certain conditions, e.g., if it is differentiable. Also called trigonometric series.

Z Z

Z

a

Similar formulas hold also for triple or multiple integrals. function One of the most important mathematical objects. In order to define a function it is necessary first of all to have two sets associated with the function called the domain and the range. Then we can formally define the function as some relation (mapping, transformation) that translates each element of the domain to exactly one element of the range. If the requirement of having exactly one corresponding element in the range is removed, then instead of function we have a relation. Functions can be represented in many different ways. 1) The first, least common form of representation is the verbal representation. As an example we can describe a function as follows: The function translates each value of the variable to its double. 2) The function can be represented also numerically, by a table, matrix, chart, etc. For example, if the domain of the function is the set {1, 2, 3, 4, 5} and the range is {6, 9, 13, 17} then the function can be given as a list of ordered pairs: {(1, 6), (2, 9), (3, 17), (4, 9), (5, 13)}, meaning that the function translates the point 1 to the point 6, point 2 to the point 9, etc. 3) The function can have graphical representation when instead of numeric values just the graph is given from where the numeric values may be found. This method of representing functions is not very accurate. 4) The most important form of representation of functions is the algebraic (often also called analytic) form, when an explicit formula is given from which the values of the function can be calculated. The functions represented algebraically, usually written with function notation, which simplifies the process of finding the values at a given point. For example, if a function given with the notation f (x) = 2x2 − 3, then the value of the function at the point x = −2.7 will be found by substituting this value for x : f (−2.7) = 2(−2.7)2 − 3 = 11.58. Functions given algebraically are further categorized as algebraic and transcendental. On their turn, alge-

56 2 2 2 2 nπx braic functions have subcategories of polynomial, raun (x, t) = e−n π a t/M sin , n ≥ 1. M tional, radical functions and the transcendental functions can be exponential, logarithmic, trigonometric The wave equation: and others. For specific examples see the corresponda2 uxx = utt , 0 < x < M, t > 0 ing entries. nπx nπat Further, functions could be increasing, decreasing, un (x, t) = sin cos , n ≥ 1. M M even, odd, one-to-one, periodic, continuous, differLaplace’s equation: entiable, integrable and of many other types. 5) Functions can be represented as series, such as uxx + uyy = 0, 0 < x < a, 0 < y < b power series or Fourier series. For example, the ranπx nπat tional function f (x) = 1/(1 − x) on interval (−1, 1) un (x, t) = sinh sin , n ≥ 1. b b could be represented as fundamental theorem of algebra If p(x) is a ∞ polynomial of degree n > 0, then the correspondX 1 = xn . ing polynomial equation has at least one solution in 1 − x n=0 the field of complex numbers. As a result of this theorem it follows that any polySimilarly, we have a Fourier series representation on nomial of degree n > 0 has exactly n roots (real or interval (0, π) complex), if we count the roots with their multiplicities. ∞ π − x X sin nx = . fundamental theorem of arithmetic Any posi2 n n=1 tive integer n could be factored in a product of powk1 k2 km 6) Functions could be represented also with integrals. ers of prime numbers: n = p1 · p2 · · · pm . This possible order of prime Laplace transform is the most common example of presentation is unique except 3 factors. Example: 2520 = 2 · 32 · 5 · 7. integral representations of functions.

The notion of the function could be extended also to the case of two, three or, more generally, n variables. Also, functions can be vectors themselves, or be vector-valued. For specific types of function see also Bessel functions, harmonic functions, implicit functions.

fundamental theorem of calculus Let f be a Riemann integrable function on the interval [a, b] and let F be the indefinite integral (antiderivative) of the function f on (a, b) : F 0 (x) = f (x). Then Z b f (x)dx = F (b) − F (a). a

fundamental matrix spaces For a matrix A, the collective name of the following four spaces: Row This theorem allows to calculate definite integrals space of A, column space of A, the nullspace of A, without elaborate process of partitioning the interand the nullspace of the transpose of the matrix AT . vals, constructing Riemann sums and passing to the limit. It also combines two main branches of calcufundamental solutions In the most general setlus: the differential calculus and integral calculus. ting, a (generalized) function φ is called the fundaExample: Since (sin x)0 = cos x, mental solution of the partial differential operator L, Z π if it satisfies the equation Lφ = δ, where δ is the Dirac cos xdx = sin π − sin 0 = 0. function. For specific partial differential equations we 0 have the following expressions for fundamental solutions: fundamental trigonometric identities The Heat conduction equation: identities 1 1 1 cot x = , sec x = , csc x = a2 uxx = ut , 0 < x < M, t > 0 tan x cos x sin x

57 and

G

cos x sin x , cot x = . cos x sin x See also Pythagorean identities, cofunction identities,addition and subtraction formulas, double-angle, half-angle formulas, sum-to-product, product-to-sum gamma function For the real positive variable x, formulas. the function defined by the formula Z ∞ tx−1 e−t dt. Γ(x) = tan x =

0

This function is the generalization of the factorial in the sense that Γ(n + 1) = n!. Gamma function can also be defined for complex values of argument. Gaussian elimination One of the most effective methods of solving systems of linear algebraic equations. The idea of the method consists of eliminating more and more variables in successive equations, and to arrive to the situation, when the last equation has only one unknown. This will allow to find the value of that variable and substitute in the previous equation that has only two variables, hence finding the value of the second unknown. Continuing this back substitution, we will be able to solve the system. During this process, only elementary row operations are performed, which guarantees, that the resulting system is equivalent to the original one, i.e., has the same solution set. Also, the procedure may be applied directly to the system of equations or the corresponding augmented matrix. We will demonstrate the second approach. Let a11 x1 + a12 x2 + · · · + a1n xn = b1 a21 x1 + a22 x2 + · · · + a2n xn = b2 ······························ an1 x1 + an2 x2 + · · · + ann xn = bn be our system of n equations with n variables x1 , x2 , · · · , xn and let a11 a21 . ..

a12 a22 .. .

... ...

a1n a2n .. .

b1 b2 .. .

an1

an2

...

ann

bn

58 be the corresponding augmented matrix. First, we column of the second row. Similarly, doing the operdivide first row by a11 and get 1 in the first position ation 3R1 + R3 → R3 will eliminate the first entry on of the first row. Next, we multiply first row by −a21 the third row. Now, the matrix has the form and add to the second row, to get zero on the first 1 −2 3 4 position of the second row. After dividing second row 0 5 −10 −5 . by a22 − a12 a21 /a1 , we will have the following format 0 −2 8 10 1 a0 0 0 . . . a b 12 1n 1 . . . a02n b02 On the next two steps we multiply second row by 1/5 0 1 . .. .. .. and the third row by -1/3 to get 1’s in the middle . . . . . column. Finally, after doing the operations −R2 + an1 an2 . . . ann bn R3 → R3 and multiplying the last row by -1/2, we Now, continuing, we can eliminate first element of arrive to 1 −2 3 4 the third row by using the first row, second element 0 1 −2 −1 . of the third row by using the second row, and then 0 0 1 2 dividing by the value of the resulting third element, we will make it one. As a final result we will find the This matrix corresponds to the system of equations matrix in upper triangular form, with all entries on the diagonal being 1’s: x − 2y + 3z = 4 1 a0 0 b01 12 . . . a1n y − 2z = −1 0 . . . a2n b02 0 1 . . .. .. . . z = 2, . . . . 0 0 0 . . . 1 bn which could be solved by back substitution method This triangular matrix corresponds to triangular system of equations, that is solved by the back substitution method. This same method works also for more general case of m × n systems but does not result in unique solution. Example: To solve the system of equations 2x + y − 4z = 3 x − 2y + 3z = 4 −3x + 4y − z = −2 we form its augmented matrix 2 1 −4 3 1 −2 3 4 . −3 4 −1 −2

and gives the solution set (4, 3, 1). Gauss-Jordan elimination The development of Gaussian elimination method. Here, the method does not stop at triangular matrix but makes it diagonal. Assume, that as a result of Gaussian elimination, our augmented matrix is reduced to the following one: 1 0 . . . 0

0

a12 1 .. .

... ...

0 0

... ...

a1n a2n

.. . an−1n 1

b1 b2

.. .

bn−1 bn

Now, if we multiply the last row by −an−1n and add it to the previous row, the nth term of that row will become zero. The only non-zero elements of that row will remain the (n − 1)st element (which is on the As the first step we interchange first and second diagonal and equals one) and the last entry, which rows to get a 1 on the left upper corner (notation: will be equal now bn−1 − an−1n bn . Next, using both R1 ↔ R2 ), then multiply first row by -2 and add to bottom rows, we can eliminate two elements on the the second row (−2R1 +R2 → R2 ) to get 0 on the left (n − 2)nd row, except the diagonal element (equals

59 one) and the (n + 1)st entry. Continuing this way up already solved for all variables. we eventually arrive to the matrix general multiplication rule for probabilities 1 0 . . . 0 b0 Let A and B be two events, not necessarily indepen1 0 dent. Then the multiplication rule for probabilities 0 1 . . . 0 b 2 . . . . generalizes as follows: . . .. .. , . . 0 0 . . . 0 b0n−1 P (A ∩ B) = P (A) · P (B|A), 0 0 . . . 1 bn where P (B|A) is the conditional probability of the which, in turn, is equivalent to the system of equa- event B given A. Another notation for this rule is tions P (A and B) = P (A) · P (B|A). x1 = b01 , x2 = b02 , · · · , xn−1 = b0n−1 , xn = bn .

general solution of linear equations For differHence, this last system provides the solution imme- ential equations. Let Lf (x) be some linear differential operator. The general solution of the equation diately. As an example we will continue solution of the system Lf (x) = g(x) is the solution that presents all possible solutions of that equation. Hence, general solution is from the previous entry. To solve the system not a function but a family of functions, depending on some numeric parameters. If the equation is of the 2x + y − 4z = 3 nth degree, then, as a rule, there are n parameters in x − 2y + 3z = 4 this family of functions. For linear equations of any degree, the general solution of the non-homogeneous −3x + 4y − z = −2 equation Lf (x) = g(x) is the sum of the general soluwe apply the Gaussian elimination method and artion of the homogeneous equation Lf (x) = 0 and any rive to the matrix particular solution of the original equation. Similar 1 −2 3 4 definition and facts are true also for systems of linear 0 1 −2 −1 . equations. 0 0 1 2 general solution of trigonometric equations Continuing by the Gauss-Jordan method, we first do The solution of any trigonometric equation that prothe operation 2R3 +R2 → R2 to eliminate third entry vides all possible answers. Example: The equation on the second row and then the operation −3R3 + cos 4x sin x = 0 R1 → R1 eliminating the third entry on the first row. The result is the matrix splits into two equations sin x = 0 and cos 4x = 0. The first one has the general solu1 −2 0 −2 0 1 0 3 . tion x = πn, n = 0, ±1, ±2... and the second one gives x = π/8 + πn/4, n = 0, ±1, ±2... All the solu0 0 1 2 tions of the original equation are described by the set On the last step we add two times row two to the S = {x|x = πn, x = π/8 + πn/4, n = 0, ±1, ±2...}. first row and finally general term An expression used in various 1 0 0 4 situations. Most commonly is used for polynomials 0 1 0 3 to indicate the term aj xj with some j = 0, 1, 2, ... 0 0 1 2 generators of the cone See the definition of the which corresponds to the system cone. x = 4,

y = 3,

z = 2,

60 geometric distribution One of the probability the gradient is a vector, which coordinates are the distributions, where the probability that k trials are partial derivatives of the function: needed to get one success is ∂f ∂f ∂f 5f (x1 , x2 , · · · , xn ) = ( , ,···, ). P (X = k) = (1 − p)k−1 p ∂x1 ∂x2 ∂xn and p is the probability of success on each trial. geometric mean For given set of n ≥ 1 nonnegative numbers, the quantity √ n a1 · a2 · · · an .

The gradient vector points to the direction of largest growth of the function. gradient vector field For functions of three variables. Let f = f (x, y, z) be a scalar function of three variables. Then the gradient of the function written in the vector form

geometric sequence or progression A sequence ∂f ∂f ∂f of real numbers (finite or infinite), where the ratio of i+ j+ k 5f (x, y, z) = each term (starting from the second) to the previous ∂x ∂y ∂z term is the same constant. The general term of these is called gradient vector field. Similar definition is sequences is given by the formula valid for functions of any number of independent variables. See also curl of a vector field, divergence of a an = a1 rn−1 , vector field. where a1 is the first term of the sequence and r is the Gram-Schmidt process A method of orthonorratio of the sequence. malization of a basis of a vector space. Suppose P∞ geometric series A series a , where the n {v , v , · · · , vn } is a basis of the vector space V . We n=1 1 2 terms form a geometric sequence. If the ratio r of start with any vector of the system, say v1 and call it that sequence satisfies |r| < 1, then the series is con- u1 To get the second vector, we take v2 and subtract vergent and the sum is given by the formula from it its projection on the vector v1 : ∞ X n=1

an =

a1 . 1−r

For an arbitrary ratio r, the partial sums of the series are given by the formula (m ≥ 1 is an arbitrary integer) m X a1 (1 − rm ) an = . 1−r n=1

u2 = v2 − proju1 v2 . This vector is orthogonal to u1 . In the next step, we take the third vector in our basis and subtract from it its projections on first two vectors: u3 = v3 − proju1 v3 − proju2 v3 .

The vector u3 is now orthogonal to both u1 and u2 . golden ratio Two numbers, a and b, form the Continuing this process, we get a new system of vecgolden ratio, if the ratio of the larger number to tors {u1 , u2 , · · · , un }, which is orthogonal. In the smaller, say a/b, is equal to the ratio of their sum last step of this process we just normalize this last system by dividing each of the vectors by their length to the larger number (a + b)/b. Simple calculations √ show that the exact value of that ratio is (1 + 5)/2, (magnitude, norm). The new system which is approximately 1.618033988... u2 un u1 w1 = , w2 = , · · · , wn = ||u || ||u || ||u 1 2 n || golden ratio rectangle A rectangle, where the length and width√form the golden ration, i.e., l/w = is now orthonormal, because each vector has length (l + w)/l = (1 + 5)/2. one. gradient For a function f of several real variables, The Gram-Schmidt process works also for infinite

61 bases. graph of an equation Let y = f (x) be some equation defined on an arbitrary interval [a, b] (finite or infinite). The graph of the equation is the geometric place of all points (x, f (x)) on the Cartesian plane, when the variable x varies over its interval of existence. Every graph represents some kind of curve, depending on the nature of the equation. Graphs are helpful visual representations of equations. graph of a function Let y = f (x) be some function defined on an arbitrary interval [a, b] (finite or infinite). The graph of the function is the geometric place of all points (x, f (x)) on the Cartesian plane, when the variable x varies over its interval of existence. Every graph represents some kind of curve, depending on the nature of the function f . The difference between the graph of a function and the graph of an equation is that for the function to each point on the interval [a, b] there is only one corresponding point on the plane. Graphs are helpful visual representations of functions. greater than An inequality, stating that one quantity is greater than the other, with notation A > B. If the quantities are real numbers, then this statement means that A is to the right of B on the number line. See also less than.

greatest lower bound Also called infimum. If a function f or a sequence an are bounded below, then ”biggest” of all possible lower bounds is the greatest lower bound. Example: The sequence an = 1+1/n is bounded below by any number M ≤ 1 but not by any number greater than 1. The number 1 is the greatest lower bound. See also least upper bound. Green’s identity Let u(x, y) and v(x, y) be two times continuously differentiable functions in the finitely simple region D with the boundary C. Denote by dA the area measure in D, by ds the arc measure on C and by n the unit outer normal to the boundary C. Then Z Z Z ∂v ∂u (u∆v − v∆u)dA = (u −v )ds. ∂n ∂n D C

greater than or equal to An inequality, stating that one quantity is greater than the other, or equal to it, with notation A ≥ B. If the quantities are real numbers, then this statement means that A is to Here ∆ denotes the Laplace operator. the right of B on the number line, or the two points Green’s identity Let M (x, y) and N (x, y) be concoincide. See also less than or equal to. tinuously differentiable functions in the finitely simgreatest common factor (GCF) For two pos- ple region D and let C be the boundary of D. Then Z Z Z itive integers m and n, the largest integer, that ∂N ∂M M dx + N dy = ( − )dxdy. divides both of that numbers, denoted by gcf (m, n). ∂y C D ∂x Example: gcf (12, 18) = 6. Also called greatest This theorem is one of the possible generalizations common divisor (GCD). of the Fundamental Theorem of Calculus to double greatest integer function For the given real integrals, because it allows to calculate integrals over number x the function [[x]] is defined to be the plane domains as line integrals over the boundaries greatest integer such that n ≤ x. This function of that domains. has jump discontinuity at each integer point and is grouping symbols The symbols ( ),[ ],{ } and constant for all values between two integers.

62

H

sometimes others, that are used to group numbers, functions, expressions, etc. Primarily are used to indicate and determine the order of operations to be performed. In many cases some other symbols also play the role of grouping symbols.√ For example, in half-angle formulas In trigonometry, the formuexpressions 5 + |3 − 6| − 4 or 3 − 4 + 5 the opera- las 1 + cos θ θ tions inside the absolute value and square root signs cos2 = should be done before any other operations. 2 2 growth rate Also could be called rate of growth. For a given variable quantity s(t) the rate with which it grows. Mathematically it is the same as the derivative, i.e., growth rate of s is s0 (t).

sin2 tan

θ 1 − cos θ = 2 2

θ 1 − cos θ sin θ = = . 2 sin θ 1 + cos θ

Similar formulas exist for other trigonometric functions but are rarely used. half-life The period of time required for a quantity (usually a radioactive material) to reduce to its half. If the initial amount of the quantity is A0 , then the amount at the time t is given by A(t) = A0 e−λt . Half-life depends on material (and is reflected in constant λ) and can vary from a few seconds to many thousand years. half-open intervals Intervals of the form [a, b) or (a, b], where a and b are any real numbers. Notation means that one of the ends of the interval is included and the other one is not. half-plane One of the four possible configurations on the Cartesian plane: {(x, y)|x > 0}, {(x, y)|x < 0}, {(x, y)|y > 0}, {(x, y)|y < 0}. More generally, the solution of a linear inequality ax + by < c (or other three alternative forms of that inequality involving the symbols >, ≥, or ≤), where a, b, c are real numbers. half space In three dimensional Euclidean space, the solution of any inequality of the form Ax + By + Cz < D (or other three alternative forms of that inequality), where A, B, C, D are real numbers. This is the same, as to say that half space is the set of all points, that lie on one side of some plane. This notion is possible to extend to higher dimensional spaces. harmonic function Any function u of two or more variables in a given domain, that satisfies Laplace’s equation. In the case of two variables the equation

63 has the form

helix. ∂ 2 u(x, y) ∂ 2 u(x, y) + = 0. ∂x2 ∂y 2

harmonic motion or simple harmonic motion. Describes the periodic movement of some object about the equilibrium point that can be given by one of the following equations: s(t) = a cos ωt, s(t) = a sin ωt. Here s is the displacement of the object at the time t, a is the amplitude of motion and ω/2π is the frequency of oscillation. harmonic series

The numeric series ∞ X 1 . n n=1

This series is divergent and according to Euler’s formula, its partial sums go to infinity as the natural logarithmic function. heat conduction equation In Partial Differential Equations: for a function of two spacial variables helix A curve in 3-dimensional space given by the x, y, and the time variable t, the equation equations ∂ 2 u(x, y, t) ∂ 2 u(x, y, t) ∂u(x, y, t) = + . ∂t ∂x2 ∂y 2 In the simpler case of one spacial variable x and time variable t the equation has the form a2 uxx = ut . Heaviside function The function n 1 if x ≥ 0 f (x) = 0 if x < 0

x = cos t, y = sin t, z = t. Here t is a real parameter. As it increases, the point (x, y, z) traces a helix about the z-axis. Heron’s formula For the area of a triangle with the sides, equal to a, b and c. Denote s = (a+b+c)/2. Then the area A is given by the formula p A = s(s − a)(s − b)(s − c).

This function plays important role in physics and electrical engineering. See also Laplace transforms higher order derivatives Since the derivative of of special functions. a function is also a function, we can calculate the helicoid A 3-dimensional surface given by the derivative of that function. If it exists, then that will equations be the second derivative of the function: x = r cos t, y = r sin t, z = αt,

f 00 (x) = (f 0 (x))0 =

d2 f . dx2

where both r and α range from −∞ to ∞. For any fixed values of that parameters, helicoid becomes a Similarly, the third derivative, f 000 (x) =

d3 f dx3 ,

the

64 4 fourth, f (4) (x) = f iv (x) = ddxf4 , and the nth derivan tive f (n) (x) = ddxnf are defined, if they exist.

The equation y 00 (t) + p(t)y 0 (t) + q(t)y(t) = 0

Hilbert space A linear vector space, equipped is homogeneous, but the equation with inner product and the notion of distance assoy 0 (t) − 2y(t) = 3x2 ciated with that inner product. The term almost exclusively applies to infinite dimensional vector spaces. is not. In this last case equation is called nonhistogram In statistics, a histogram is a graphical homogeneous. display of frequencies. A histogram is the graphi- Homogeneous equations are also classified by their cal version of a table which shows what proportion order (first, second,etc.), by the type of coefficients of cases fall into each of several specified categories. (constant or variable), by linearity and so on. In The histogram differs from a bar chart in that it is cases when we have several equations with several the area of the bar that denotes the value, not the unknown functions, we get system of homogeneous height. The categories are usually specified as non- equations. overlapping intervals of some variable. Another dif- Hooke’s law In physics, this law states that the ference between histograms and bar graphs is that force necessary to hold a spring stretched x units bethe bars in histograms are adjacent, not separated as yond the equilibrium point is proportional to the disfor bar graphs. placement x, if its small. Symbolically, f (x) = kx, where k is a positive number called spring constant. horizontal asymptote See asymptote. horizontal axis The x-axis in Cartesian coordinate system. It is given by the equation y = 0. horizontal line A line, that is parallel to the xaxis in Cartesian coordinate system. Any horizontal line is given by the equation y = a, where a is any real number.

homogeneous algebraic equations Systems of linear algebraic equations, where all of the right sides are zero. Homogeneous systems are always consistent, because they always have the trivial solution (i.e. all zero solution). Homogeneous systems may have also non-trivial solutions, if any of the equations is a linear combination of multiple other equations.

horizontal line test If a function f (x) is given graphically, and any horizontal line intersects that graph only once, then the function is one-to-one. If even at one point this condition is violated then the function cannot be one-to-one. This test allows to determine if the given function has an inverse or not but is not accurate.

hyperbola One of the three main conic sections. Geometrically, a hyperbola is the location (locus) of all points in a plane, the difference of whose distances from two fixed points in the plane is a positive constant. The two points, from which we measure distances, are called foci (plural for focus). Equivahomogeneous differential equation A differen- lently, the hyperbola could be described as the result tial equation where only the unknown function and of cutting double cone by a plane parallel to its axis. its derivatives are present, possibly multiplied by Hyperbola is a smooth curve with two branches. Alsome (variable or constant) coefficients. Examples: ternative geometric definition could be given with the

65 use of eccentricity. See corresponding definition. Algebraically, the general equation of a hyperbola is given by the equation Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0, where A, B, C, D, E, F are real constants and A · C < 0. In the case, when the foci are located on one of the coordinate axes and center coincides with the origin, this equation could be transformed into standard form y2 x2 − = 1, a2 b2 where √ the constants a, b have special meaning: c = a2 − b2 is the distance from origin to a focus. hyperbolic functions The functions Moreover, the intersection points of hyperbola with one of the axes are called vertices of hyperbola and ex − e−x ex + e−x sinh x = , cosh x = , their distance from origin is equal to a. The line 2 2 connecting two foci is called transverse axis and ex + e−x ex − e−x the perpendicular line (passing trough origin) is the , coth x = x , tanh x = x −x conjugate axis. Each hyperbola of this form has e +e e − e−x two slant asymptotes given by the formulas y = ± ab x. Hyperbolic secant and cosecant functions are defined as reciprocals of hyperbolic cosine and sine functions, In the more general case,when the center of hyperbola respectively. The sinh x and cosh x functions are the is located at some point (h, k) but the transverse and most used with a very limited use for other four hyconjugate axes still parallel to coordinate axes, the perbolic functions. standard equation becomes hyperbolic identities Identities for hyperbolic functions, analogous to corresponding trigonometric (y − k)2 (x − h)2 − = 1, 2 2 identities. Here the equivalent of the Pythagorean a b identity is the following: and the equations of the slant asymptotes are y = cosh2 x − sinh2 x = 1. k ± ab (x − h). In all cases the eccentricity is given by the formula e = c/a. Both of the above cases happen when in the general Another example is the formula equation the term Bxy is missing. In the case B 6= 0 sinh(x + y) = sinh x cosh y + cosh x sinh y. the result is still a hyperbola which is the result of rotation of one of the previous simpler cases about hyperbolic paraboloid The three dimensional the center of hyperbola. surface given by the formula The hyperbola could also be given by its polar equation: x2 y2 z − 2 = . 2 a b c de de r= or r = , hyperbolic substitution A method of calculation 1 ± e cos θ 1 ± e sin θ of indefinite integrals, when the independent variable where d > 0 and e > 1 is the eccentricity of the is substituted by one of the hyperbolic functions. hyperbola. Substitutions are used to evaluate indefinite integrals

66 when ready to use formulas are not available. See also trigonometric substitutions and substitution method.

hypersphere Usually, the name of the ndimensional sphere given by the equation x21 + x22 + · · · + x2n = r2 , where r is the radius. hypervolume Usually, volume in n-dimensional Euclidean space.

hyperboloid (1) Hyperboloid of one sheet is given by the formula

x2 y2 z2 + − =1 a2 b2 c2

and represents a surface that is connected. (2) Hyperboloid of two sheets is given by

x2 y2 z2 − − =1 a2 b2 c2

and consists of two pieces.

hypocycloid Geometrically, a curve that is traced by a point on a circle that rolls inside a bigger circle. Depending on ratio of radiuses of the circles, the result is a different curve. If that ratio is 1/4, the result is called astroid (see picture above). In general, a hypercycloid with n + 1 cusps is given by parametric

67 equations

I

1 1 x = cos θ + cos nθ, y = sin θ − sin nθ. n n hypotenuse In the right triangle, the side, opposite to the right angle. The other two sides are called legs.

identity (1) An equality, containing constants and/or variables, that is true for all values of the variables, whenever the expression makes sense. Examples are: (a + b)2 = a2 + 2ab + b2 , sin2 θ + cos2 θ = 1, 1 + tan2 x = sec2 x. The first two identities are valid for all values of a, b or θ, and the last identity is valid for all values of x 6= π/2 + πk, k = 0, ±1, ±2, ..., because the functions involved in the identity are not defined for that values of argument. See also trigonometric identities. (2) A number, that added to other numbers or multiplied by other numbers does not change them. See additive identity, multiplicative identity.

hypothesis A statement or proposition, which validity is not known but which is very likely to be true based on experimental or theoretical observations. One of the best known is the so-called Riemann hypothesis.

identities, identities.

trigonometric See

trigonometric

identity matrix The square matrix where all the entries are zero except of the main diagonal, where all the entries are ones. Example: hypothesis testing In Statistics. the procedure of 1 0 0 testing a working assuption (hypothesis) using a set I = 0 1 0 of values from a sample. For details see null hypoth0 0 1 esis and alternative hypothesis. When we multiply the identity matrix by any other matrix from the right or left, the given matrix does not change: IA = AI = A. identity function The function f (x) = x which leaves each point in its place. identity transformation A transformation T of a vector space V that leaves all elements of the space unchanged: For any v ∈ V , T v = v. Also called identity operator. image of a linear transformation. If T : V → W is a linear transformation from one vector space to another, then for any vector u ∈ V the vector T (u) ∈ W is its image. The set of all images is called the image of the transformation T . imaginary axis The vertical axis in the complex plane. When the complex plane is identified with the

68 Cartesian plane, imaginary axis is the same as y-axis. contains a point (a, b). Assume that F (a, b) = 0, ∂F ∂y (a, b) 6= 0, and both partial derivatives of imaginary number A number√of the form ai, F are continuous on the disk. Then the equation where a is a real number and i = −1 is the imagiF (x, y) = 0 defines y as an explicit function of x near nary unit. the point (a, b) and the derivative of that function is √ imaginary unit The number i = −1. This num- given by the formula ber is not real, because i2 = −1. See also complex dy ∂F ∂F numbers. =− / . dx ∂x ∂y implicit differentiation When a function is given implicitly (see implicit function), then it is still pos- Similar statements are valid for functions of three or sible to calculate the derivative of that function. Let more variables also. the function is given by the equation F (x, y) = 0 and assume that F has partial derivatives with respect to implicit solution A solution of an (usually differboth variables. In that case, we will be able to cal- ential) equation which is given as an implicit function culate the derivative of y with respect to the variable F (x, y) = 0. x if we view y as a function of x and use the chain implied domain When the domain of some funcrule. Example: If the implicit function is given by tion is not stated explicitly but is clear from the defithe equation nition of the function, then it is said that the domain arctan y/x = x, is implied. then differentiation by x gives dy dx x − y x2 + y 2

=1

improper fraction A fraction, where the numerator is greater than or is equal to the denominator. Examples: 57 , 25 25 .

improper integral An integral, where either one or both integration limits are infinite, or the function is unbounded on the bounded interval. dy x2 + y 2 + y (1) Let the function f (x) be defined on the interval = , RA dx x [a, ∞) and assume that the integral a f (x)dx exists for any given number A ≥ a. Then the limit of that which itself is an implicit function. integral (finite or infinite) is the integral of f over the implicit function A function of two variables, interval [a, ∞): that is not explicitly solved for any of the variZ ∞ Z A ables. This relation can be expressed as an equation f (x)dx = lim f (x)dx. F (x, y) = 0 with some function F . Examples are A→∞ a a x2 + y 2 − 1 = 0, 2 sin3 x − 3xy = 0. Similar definition holds for functions of several variables too. Let In case this limit is finite, the function is called intex1 , · · · , xn , y be (n + 1) variables and F represents grable on the given interval. The improper integrals some functions of that variables. Then the equa- over the intervals (−∞, b] and (−∞, ∞) are defined tion F (x1 , · · · , xn , y) = 0 is an implicit function if similarly. the equation is not solved for any of the variables. (2) Let f (x) be defined on a bounded interval [a, b]

and, after the simplification,

Implicit Function Theorem gives conditions under which a function defined implicitly, can be solved for the dependent variable. We give the simplest version for the case of two variables. Let F (x, y) = 0 be an implicit function defined on some disk, that

but be unbounded there, for example, as we approach the right endpoint b. Assume also, that on each interval [a, b − δ] the function is bounded and the integral R b−δ f (x)dx exists. Then the limit of that integral a (finite or infinite) is the integral of f over the inter-

69 val [a, b]:

is inconsistent because it has no solutions.

increasing/decreasing test Also called increasing/decreasing theorem. δ→0 a a (1) If the function is defined on some interval I and 0 In case the limit is finite, the function is called in- f (x) > 0 for all points on that interval, then the function is increasing. tegrable on the given interval. Improper integrals, (2) If the function is defined on some interval I and when the left end-point or some inside point of the 0 f (x) < 0 for all points on that interval, then the interval are the points where f is not bounded, are function is decreasing. defined in a similar manner. Z

b

Z

f (x)dx = lim

b−δ

f (x)dx.

improper integral comparison theorem For improper integrals on infinite intervals. Assume that f and g are continuous functions defined on some interval [a, ∞) and f (x) ≥ g(x) on that interval. R R∞ ∞ (1) If f (x)dx is convergent, then a g(x) dx a is convergent; R∞ R∞ (2) If g(x)dx is divergent, then a f (x) dx a is divergent. impulse function Let g(t) be a function defined for all real values t, with the condition that it is zero outside some ”small” interval [t0 − τ, t0 + τ ] and with finite integral on that interval. These kind of functions represent some type of impulse (mechanical, R∞ electrical, or other) and its integral I(τ ) = −∞ g(t)dt is the total impulse of the force g(t). The limiting case of these type of impulse functions as the interval of its existence goes to zero, is the Dirac function.

increasing function The function f (x) defined on some interval I is increasing, if for any two points x1 , x2 ∈ I, x1 < x2 , we have f (x1 ) < f (x2 ). increasing sequence A sequence of real numbers {an }, n ≥ 1, is increasing, if am > ak whenever m > k. indefinite integral For a given function f (x) on some interval, indefinite integral is any function F (x), which derivative is that function: F 0 (x) = f (x).

In fact, indefinite integral is not one specific function, but a family of functions. Along with F , any function of the form F (x) + C, where C is an arbitrary constant, is also indefinite integral of f . For indefinite integral of a function f we have the notation R f (x)dx. Examples: Z sin xdx = − cos x + C, impulse response Assume we have some differenZ tial equation. We can take the equation p dx √ = ln(x + a2 + x2 ) + C. 2 2 a +x ay 00 + by 0 + c = g(t) Indefinite integral is also called antiderivative. for an example. If in the right-hand side the function indefinite matrix A square matrix for which the g is the Dirac function δ(t), then the solution of this expression xT Ax takes both positive and negative equation is called impulse response (to the impulse values. Here x is an arbitrary vector and xT is its δ). transpose. See also positive definite, negative defiinconsistent linear system A system of linear nite matrices. equations that has no solution. The system is consisindependence of path Let F be some vector field tent, if it has at least one set of solutions. Example: in two or three dimensional space and assume that The system of equations some piecewise smooth curve (path) C is given by the vector-function r(t). The integral x − 2y = 5 Z F · dr 2x − 4y = 3 C

70 is said to be independent of path, if its value depends on initial and terminal points R of the path C only. Theorem 1. The integral C F · dr is independent of R path if and only if for any closed path C, C F·dr = 0. Additionally, the next theorem establishes when this situation takes place. Theorem 2. Suppose F is a continuous R vector field in some open connected region D. If C F · dr is independent of path in D, then F is a conservative vector field, i.e., there exists a function f such that F = 5f.

summation notation) presentation of a finite or infinite series m X an n=1

the parameter n. As a rule, index of summation takes either integer or whole values.

indicial equation An algebraic equation, associated with some differential equations. This equations come out in cases when we solve differential equation independent events Two or more events are in- by series method around regular singular point. Exdependent, if none of them affects the outcome of any ample: Assume we have the equation of the others. If A and B are the two events and P 2x2 y 00 − xy 0 + (1 + x)y = 0. denotes the probability, then the relation of independence could be expressed in the form P (A and B) = The point x = 0 is a regular singular point and to P (A)P (B), or, equivalently, P (A ∩ B) = P (A)P (B). solve this equation we need to solve corresponding Independence could be extended also to three or more indicial equation events and the definition will be similar. 2r(r − 1) − r + 1 = 0, independent observations For an observation independence has the same meaning as for an event. which produces two roots r = 1, 1/2. Now, two Similarly, when collecting samples we get indepen- solutions of the original equation are found by muldent samples if no one depends on any other collected tiplying xr by certain power series: sample. See also simple random sample. # " ∞ X (−1)n 2n xn independent variable In an equation y = f (x) , y1 = x 1 + (2n + 1)! the variable x. The variable y is called depenn=1 dent variable. A function of several variables " # ∞ X y = f (x1 , x2 , · · · , xn ) has n independent variables (−1)n 2n xn 1/2 y2 = x 1+ . x1 , x2 , · · · , xn . (2n)! n=1 indeterminate forms of limits Let f (x), g(x) be two functions defined around the point x = c (but not necessarily at that point). When calculating the limit f (x) lim x→c g(x)

it is possible that limx→c f (x), g(x) = 0 or limx→c f (x), g(x) = ∞. In these cases we say that the limit is indeterminate form 0/0 or ∞/∞ respectively. Also we may have 0 · ∞ indeterminate form while calculating the limit of the product of f and g. In other situations, indeterminate forms ∞ − ∞, 1∞ and 00 also may arise. In many cases these limits could be calculated by the l’Hospital rule. index of summation

induction

See mathematical induction.

induction hypothesis One of the steps in the method of mathematical induction, when we assume that certain statement is true for the value of the natural number k and then proceed to prove the statement for the next integer k + 1. inequality A statement that one quantity does not equal another quantity. The statement could be written in one of the following four possible forms: A < B, A ≤ B, A > B, or A ≥ B. Se also quadratic inequality. inference, statistical See statistical inference.

In the sigma notation (or infinite discontinuity

A function is said to have

71 infinite discontinuity at some point x = c, if one or both of the left-hand or right-hand limits is either ∞ or −∞. The function may or may not be defined at that point. infinite interval An interval, where one or both of the endpoints are infinite. The only three possible types of infinite intervals are (−∞, a), (b, ∞) and (−∞, ∞).

inflection point Let y = f (x) be a continuous function on some interval. A point P on its graph is called inflection point, if the graph changes its concavity when we pass through that point. The function y = x3 has inflection point at x = 0, because to the left of that point it is concave down and to the right it is concave up.

influential observation Or influential point. In a scatterplot one (or some) points are called influential, infinite limit (1) A sequence of numbers has inif removing them results in a very different regression finite limit if the members of the sequence grow inline. Usually, all the points that are very far from the definitely. In precise definition, {an }, n = 1, 2, 3, ..., majority of points in either horizontal or vertical dihas infinite limit, if for any given positive number M rection (or both) are influential. there exists an integer N > 1, such that an ≥ M whenever n ≥ N . initial conditions For ordinary differential equa(2) A function f (x) has infinite limit at some point tions. The conditions at the initial point to assure x = c, if for any positive M , there exist δ > 0 such that the equation has unique solution. that |f (x)| > M whenever |x − c| < δ. This last definitial point If the ordinary differential equation is inition requires modification if the point c is infinite. given on some interval [a, b] (most commonly b = ∞), In that case the definition is very similar to the defithen the point a is called initial point. nition of infinite sequence. initial point of vector A vector defined as diinfinite non-repeating decimal A decimal reprected line interval, has two endpoints on this interresentation of a real number, where no one digit val. The point where the vector starts, is the initial or group of digits repeat. This kind of decimal point. Because only the direction and length are imnumbers cannot represent integers or rational numportant for a vector, we place the initial point at the bers, so all of them are irrational. Well known exorigin of the coordinate system. amples are the decimal approximations of numbers √ 2 = 1.414213562... and π = 3.141592654.... initial side of angle Of two rays forming an angle in standard position the one that coincides with the infinite repeating decimal A decimal represenpositive x-axis. See also angle. tation of a real number, where one digit or a group of digits repeat over and over again. Each repeating initial value problem Also called Cauchy probinfinite decimal represents a rational number. The lem. For ordinary differential equations. The equaopposite is also true: each rational number can be tion along with conditions on solution at the initial represented as a finite or repeating infinite decimal. point. To solve initial value problem means to find We use a bar over the group of repeating digits to solution to the equation that also satisfies given conindicate the fact that they are repeating. Examples: ditions at the initial point. Initial conditions are cho1 2 sen to assure uniqueness of the solution. Example: 3 = 0.333.... = 0.3, 7 = 0.285714. The equation infinite sequence A sequence {an }, where the index n takes unbounded number of values (usually, y 00 + p(t)y 0 + q(t)y = 0 whole numbers or integers). for 0 ≤ t < ∞, with conditions y(0) = 0 and y 0 (0) = 1 infinite series A series where infinitely many is an initial value problem. terms are added. See also convergent series and divergent series. inner product for vectors in real vector space is defined as a real number satisfying the following con-

72 ditions: Let x, y be two vectors. Then (1) hx, yi = hy, xi; (2) For any real a, hax, yi = ahx, yi; (3) hx, xi ≥ 0 and hx, xi = 0 if and only if x = 0. Inner product is also called scalar product. In the special case when the space is the Euclidean space, and the inner product is defined as the product of coordinates of the vectors, then it is also called dot product. Inner product for complex vector space is defined in a similar manner but requires involvement of complex numbers and some additional conditions.

and indefinite integral for basic definitions. For integrals in other settings see also double integral, line integral, surface integral and many other definitions related to integrals and integration. integral curves The graphical presentations of general solutions of differential equations, that depend on one or more integration constants.

integral equation An equation where the unknown function appears under the integral sign. Two most important types of integral equations are Z b For solutions of diff (x) = K(x, t)φ(t)dt

instability of critical point ferential equations. Let y dy =r 1− y dt K

a

and Z

b

φ(x) = f (x) + λ

K(x, t)φ(t)dt. a be the logistic equation. The functions y = K and y = 0 are two constant solutions of this equation. Here f and K are given known functions and φ is the The other solutions of this equation with the initial unknown one. These equations are called Fredholm equations of the first and second type respectively. condition y(0) = y0 are given by the formula integral test Gives convergence criteria for positive series. Assume we can find a positive, continuous, and decreasing function defined onPinterval The first solution y = K is stable because all the so- [1, ∞), such that an = f (n). Then the series ∞ n=1 an R lutions (1) approach that solution as t → ∞. On the is convergent if and only if the integral ∞ f (x)dx is 1 other hand the solution y = 0 is unstable because the convergent. solutions (1) do not approach it. This solution is also integral transform A transform which is given called unstable critical point. with an integral formula. An integral transform has instantaneous rate of change If a quantity is the form changing during the time period [t1 , t2 ], then the avZ b erage rate of change is its net change divided by the T f (x) = K(x, t)f (t)dt, a elapsed time t2 − t1 . When the interval [t1 , t2 ] is getting smaller and smaller (eventually becoming a where a and b are any numbers (including infinity) single point), the resulting value is the instantaneous and K(x, t) is some (usually integrable) function of rate of change of the quantity. This is one of the main two variables, called kernel of integral transform. In problems that helped to create differential calculus. this formula the function f is regarded as a variable y=

y0 K . y0 + (K − y0 )e−rt

instantaneous velocity change for velocity.

(1)

The instantaneous rate of

or ”vector” and integral transforms are always linear. One of the most common integral transforms is the Laplace transform.

integers The set of integers is the union of natural (or counting) numbers {1, 2, 3, · · ·}, their opposites integrand The function which is getting integrated. In expression {−1, −2, −3, · · ·} and the number 0. Z 2 integral One of the most important notions of cale−x dx culus. Has two main meanings. See definite integral 2 + x2

73 2

the function f (x) =

e−x 2+x2

= 2 ln |x − 1| − ln |x − 2| + C.

is the integrand.

integrating factor Assume we have a general first order linear differential equation

Z

12 dx x4 − x3 − 2x2

0

y + p(t)y = g(t) Z with some given functions p and g. In order to find the general solution of this equation, we multiply all terms by some positive function µ(t), called integrating factor, which makes the left side equal to the derivative of the function y(x)µ(x). This R function can be chosen to be equal µ(t) = exp p(t)dt and the general solution of the equation is given now in the form Z t 1 µ(s)g(s)ds + c . y= µ(t) t0 integration The procedure of calculating definite or indefinite integrals. Unlike differentiation, where the derivative of any analytically given function could be found by applying differentiation rules, integration process is much harder and for many functions there are no simple indefinite integrals. For different methods of finding indefinite integrals see integration by partial fractions, integration by parts, integration by substitution, by trigonometric substitution, termby-term integration. For approximate integration of definite integrals see approximate integration, numerical integration, Simpson’s rule, trapezoidal rule.

=

3 1 6 4 − 2+ − x x x−2 x+1

= 3 ln |x| +

2 =2

dx

6 + ln |x − 2| − 4 ln |x + 1| + C. x Z

Z

x3 + 1 dx (x2 + 1)2

x x 1 − + 2 x2 + 1 (x2 + 1)2 (x + 1)2

= ln(x2 + 1) +

dx

x 1 + + tan−1 x + C. x2 + 1 x2 + 1

integration by parts One of the integration methods. Let u and v be differentiable functions. Then Z Z udv = uv − vdu and the identical formula holds for the definite integrals: Z b Z b b udv = uv|a − vdu. a

a

integration by partial fractions A method of This formula allows to calculate many integrals that integration of rational functions. Let f (x) = otherwise were impossible to handle. Example: With P (x)/Q(x) be a proper rational function. Then it u = ln x, v = x, could be decomposed into the sum of partial fracZ Z tions of one of the following forms: ln x dx = x ln x − dx = x ln x − x + C. A Ax + B , , (ax + b)j (ax2 + bx + c)k integration by substitution One of the integration methods. Let f (x) and u = g(x) be two funcwhere in the second case the denominator is irre- tions an u be also differentiable. Then ducible over real numbers. Now, integration of the R Z Z integral f (x)dx is reduced to integration of simpler 0 f (g(x))g (x) dx = f (u) du. fractions. Examples: Z Z This formula allows to calculate certain integrals, x−3 2 1 dx = − dx 2 that have special form. Example: To calculate the x − 3x + 2 x−1 x−2

74 R√ integral 2x + 1 dx, we notice, that with substitution u = 2x + 1, du = 2dx, and Z Z √ 1 √ 2x + 1 dx = udu 2 1 3/2 1√ 2x + 1 + C. u +C = 3 3 integration by trigonometric substitution A method of evaluating integrals containing expressions √ of the type a2 ± x2 . In that case the substitutions x = a sin θ, x = a tan θ or x = a sec θ often result in simpler integrals to evaluate. Examples: (1) To evaluate the integral Z dx √ , ( 1 − x2 )3 =

find the y-intercept of the function y = x2 − 4x + 3, we plug-in x = 0 and find y = 3 and the point is (0, 6). To find the x-intercepts, we plug-in y = 0 and solve the equation x2 − 4x + 3 = 0, which has solutions x = 1, 3 and the intercepts are (1, 0), (3, 0). intermediate value theorem Let f (x) be a continuous function on the closed interval [a, b]. Assume that L is a number between the values f (a) and f (b) of the function at the endpoints. Then there exists at least one point c such that a ≤ c ≤ b and f (c) = L. This theorem helps sometimes to find the x-intercepts of functions. For example, if f (a) < 0 and f (b) > 0 then the theorem means that there is at least one xintercept inside the interval [a, b].

interpolation If some points are given on the plane, then any curve passing through that points we substitute x = sin θ, dx = cos θdθ and get is called interpolating curve. If that curve repZ Z resents the graph of some polynomial, then it dx cos θdθ √ = = tan θ + C, 3 is called interpolating polynomial. In the most 2 3 (cos θ) ( 1−x ) general case we may fix some points of the and, returning to original variable, the answer transgiven function f creating a set of ordered pairs √ fers to x/ 1 − x2 + C. {(x1 , f (x1 )), (x2 , f (x2 )), ..., (xn , f (xn ))} and the inR 1 (2) To evaluate the integral √4+x dx, we use the terpolating polynomial p(x) will represent an approx2 2 substitution x = 2 tan θ dx = 2 sec θdθ and see that imation to the function f which additionally coincides with given function at all prescribed points. the integral is transformed to Depending on the degree of the polynomial the inZ terpolation will be linear, quadratic, cubic, etc. See sec θdθ = ln | sec θ + tan θ| + C also extrapolation. ! √ intersection A point where two curves coincide. 4 + x2 x = ln + + C. If the curves are given by equations y = f (x) and 2 2 y = g(x), then they intersect, if for some point integro-differential equation An equation, x = c, f (c) = g(c). When a curve intersects with one where the unknown function is both integrated and of the coordinate axes, then the intersection point is differentiated. Example: called intercept. Z b intersection of sets If A and B are two sets of φ00 (x) = f (x) + K(x, t)φ(t)dt, some objects, then their intersection A ∩ B is dea fined to be the set of all elements that belong to where f and K are given functions and φ is to be both A and B. Example: If A = {1, 2, 3, 4, 5, 7} and determined from this equation. B = {2, 4, 6, 8} then A ∩ B = {2, 4}. In case when two sets have no common elements we say that their intercepts The intersections of the graph of a intersection is the empty set. function with the coordinate axes. For y = f (x), the y-intercept is the point, when x = 0 and the interval A subset of the real line with the propx-intercept(s) is the point, when y = 0. Example: To erty that for any two points in that subset, all the

75 √ points between that two are also included. Intervals 3 x; f (x) = 2x , f −1 (x) = log2 x. could be closed (include both endpoints), open (do inverse Laplace transform The transformation, not include endpoints), or half-open (include only one that translates the Laplace transform of some funcof endpoints). The four possible configurations are: tion back to the same function. Symbolically, [a, b], (a, b), [a, b), (a, b]. In this definition one or both endpoints could be infinite. See infinite interL−1 Lf (t) = f (t). val. The inverse Laplace transform exists under certain interval of convergence For a power series conditions on the original function and is unique. Analytically it is given by a formula involving complex ∞ X n variable integration. A list of inverse Laplace formuan (x − c) las could be deduced from direct Laplace transform n=0 formulas working backwards. See the corresponding the interval (c − r, c + r), where the seris converges. entry. The number r is called radius of convergence and may be zero, some positive number, or infinity. Some se- inverse matrix For a given square matrix A the ries also converge at the endpoints, hence the interval matrix B, that satisfies the conditions AB = BA = I, where I is the identity matrix. Inverse matrix is becomes closed. Example: The series denoted by A−1 and it exists if and only if the deter∞ X minant of the matrix A is not zero. There is a simple xn formula for calculation of inverses of 2 × 2 matrices. n n=1 If a b A= has the half-open interval [−1, 1) as its interval of conc d vergence, because when x = 1, it results in diverging harmonic series and for x = −1 it is the converging is the given matrix, then alternating harmonic series. 1 d −b . A−1 = ad − bc −c a invariant Something, that does not change when certain transformation is performed. Examples: For For general n × n matrices the inversion formula uses the function f (x) = x2 the point x = 1 is invariant, cofactors and is given by because f (1) = 1. For any linear transformation T in 1 any vector space V , the origin is invariant, because (Cij )T , A−1 = det(A) T (0) = 0. In a more general setting, the area of a plane geometric figure is invariant when we transform where (C )T is the transpose of the matrix where eleij this figure using only translation, rotation and reflec- ment on ij−th position is the corresponding cofactor. tion operations. This formula is very hard to use in practice, so ininverse function For function f (x) of one real stead a version of Gauss-Jordan elimination method variable with domain D, the function g(x) is its in- is used. Example: If verse, if for all appropriate values of the variable 1 2 2 A = 2 3 3, g(f (x)) = f (g(x)) = x. 3 4 5 The inverse of the function f is denoted by f −1 . we form a 3 × 6 matrix by adjoining the identity maUsual sufficient condition for existence of the in- trix to A: verse is the condition of being one-to-one. Fa1 2 2 1 0 0 2 3 3 0 1 0 miliar examples of pairs of inverses are: f (x) = 2x, f −1 (x) = 1/2x; f (x) = x3 , f −1 (x) = 3 4 5 0 0 1

76 and apply Gauss-Jordan method. The result will be irrational number Any real number that is not rational. Irrational numbers cannot be written another 3 × 6 matrix as fractions, as terminating decimals or as non1 0 0 −3 2 0 terminating, repeating decimals. Common examples √ 0 1 0 1 1 −1 of irrational numbers are 2, π, e. 0 0 1 1 −2 1 irreducible polynomial The polynomial is irreand the right half of this matrix is the inverse of the ducible over a number field, if it cannot be factored given matrix A: into linear factors with coefficients from that field. A polynomial which is irreducible over some number −3 2 0 field, may be reducible over other number field. A−1 = 1 1 −1 . (1) The polynomial x2 − 2 is irreducible over the field 1 −2 1 of rational numbers, but it is reducible √ over the √ field 2 2)(x + 2). of real numbers because x − 2 = (x − inverse transformation Let T : V → W be a 2 linear transformation from one linear vector space (2) The polynomial x +4 is irreducible over rationals to another. The inverse of this transformation, de- and reals but it is reducible over complex numbers 2 noted by T −1 , is another linear transformation (if because x + 4 = (x + 2i)(x − 2i). By the Fundamenit exists), that transforms W to V and satisfies the tal Theorem of Algebra, any polynomial with rational conditions T ◦ T −1 = T −1 ◦ T = I, where I is the coefficients is reducible over the field of complex numidentity transformation and ”◦” denotes composition bers. of transformations. Sufficient condition of existence irregular singular point For differential equaof the inverse transformation is that T be one-to-one tions. Consider the equation transformation. inverse trigonometric functions See arcsine, arccosine and arctangent functions. inverse variation The name of many possible relations between two variables. The most common are:(1) y varies inversely with x means that there is a real number k 6= 0, such that y = k/x; (2) y varies inversely with x2 means y = k/x2 ; (3) y varies inversely with x3 means y = k/x3 . There are many other possibilities but rarely used. Compare also with direct variation and joint variation. inversion (1) The procedure of finding the inverse of a given object: number, function, matrix, etc. (2) In a permutation, whenever a larger number precedes a smaller one, we call it an inversion. See also transposition.

P (x)y 00 + Q(x)y 0 + R(x)y = 0 and its power series solutions. If for the point x = x0 , P (x0 ) 6= 0, then that point is called ordinary and series solution is possible near that point. If P (x0 ) = 0, then the point is singular. Additionally, the series solution is still possible if the singular point is regular, which means that the limits lim (x − x0 )

x→x0

Q(x) P (x)

and

lim (x − x0 )2

x→x0

R(x) P (x)

are finite. If any of these conditions is violated, then the point x0 is called irregular singular point. irrotational vector field A vector field F is called irrotational at some point P , if at that point curlF = 0.

invertible matrix A matrix such that the inverse isobars A contour line of equal or constant presmatrix exists. sure on a graph. involute A parametric curve given by the equaisosceles triangle A triangle that has two equal tions sides. Isosceles triangles necessarily also have two x = r(cos θ + θ sin θ), y = r(sin θ − θ cos θ). equal angles.

77 iterate In the process of successive approximation, the result on each intermediate step is called iterate.

J

iterated integral Let f (x, y) be a function defined on some rectangle R = [a, b] × [c, d]. We may integrate this function with respect of the second variable Jacobian Assume that the variables x, y, z are expressed with the help of other three variables and get a new function of one variable u, v, w and the change of variable functions x = Z d x(u, v, w), y = y(u, v, w), z = z(u, v, w) are continuf (x, y)dy. F (x) = ously differentiable. Then the matrix c

∂x ∂u ∂y J = ∂u ∂z ∂u

Now, if we integrate this resulting function with rexpect to the first variable, then we will get # Z Z "Z b

b

d

F (x)dx = a

f (x, y)dy dx. a

c

The integral on the right side is called iterated integral. Similar definitions hold for functions of three or more variables. Fubini’s theorem uses iterated integral in calculations of multiple integrals.

∂x ∂v ∂y ∂v ∂z ∂v

∂x ∂w ∂y ∂w ∂z ∂w

is called the Jacobian matrix of the transformation. The importance of this matrix is that the determinant of J plays important role in calculating integrals by change of variable. The following theorem holds: Let the function f (x, y, z) be defined and integrable in some region G and let the change of variable formulas x = x(u, v, w), y = y(u, v, w), z = z(u, v, w) transform G to some other region G0 . Then Z Z Z f (x, y, z) dxdydz G

Z Z Z =

f (u, v, w) det(J) dudvdw. G0

joint variation The name of many possible relations between three or more variables. The most common are:(1) z varies directly with x and y means that there is a real number k 6= 0, such that z = kxy; (2) z varies directly with x and inversely with y means z = k xy ; (3) z varies directly with x2 and inversely 2

with y means z = k xy . There are many other possibilities including more variables. See also inverse variation and direct variation. Jordan curve Let I = [a, b] be an interval and the functions x = f (t) and y = g(t) are defined on that interval. These equations define a plane parametric curve. This curve is called Jordan curve if: (1) The functions f, g are continuous; (2) the curve is closed, meaning that f (a) = f (b), g(a) = g(b); (3) The curve

78 is simple, i.e. f (t1 ) 6= f (t2 ) except the endpoints, and similarly g(t1 ) 6= g(t2 ). Jordan curves in the space are defined in the same way.

K

Jordan decomposition trix.

Kepler’s laws of planetary motion: (1) Every planet revolves around the Sun in an elliptical orbit with the Sun at one focus. (2) The line joining the Sun and a planet sweeps out equal areas in equal time. (3) The square of the period of revolution of a planet is proportional to the cube of the length of the major axis of its orbit.

See Jordan form of ma-

Jordan domain A plane region that is bounded by a Jordan curve. Jordan form of matrix A block-diagonal square matrix, that could be written in the form J1 0 . ..

0

0 J2 .. .

... ...

0 0 , .. .

0

...

Jn

kernel of convolution In the convolution integral Z ∞ (K ∗ f )(x) = f (t)K(x − t)dt −∞

where each Jk is itself a square matrix. The diagonal elements of this block-matrix are all the same, the sub-diagonal consists of all 1’s, and all the other elements of the matrix are zeros: λk 0 ... 0 0 1 λk . . . 0 0 0 1 ... 0 0 J = . . . . .. .. .. .. 0 0 . . . 1 λk

the function K, if we consider f as a parameter. kernel of integral transform In the integral transform Z b T f (x) = K(x, t)f (t)dt, a

the function K(x, y). Transforms vary, depending on the nature of the kernel. We get the most common Fourier and Laplace transforms, when the kernel is equal to eixy and e−xy respectively.

The entries on diagonal are the eigenvalues of the makernel of linear transformation Let T be a lintrix. These type of matrices are called simple Jordan ear transformation from one vector space V to anmatrix. Every square matrix is similar to a matrix other, W . The set of all vectors x ∈ V , such that of the Jordan form. T x = 0 is the kernel of the transformation. The kerjump discontinuity A function f (x) is said to nel is in fact a subspace of V : if u, v belong to the kerhave jump disconinuity at some point x = c, if both nel then for any constants α and β the vector αu+βv left-hand and right-hand limits of the function at that also belongs to the kernel. The dimension of the kerpoint exist, but are not equal: nel is called nullity. Kronecker’s symbol Also called Kronecker delta. A function of two discrete parameters i, j, given by the formula The Heaviside function is an example of a function 1 if i = j with jump discontinuity at point 0. δij = . 0 if i 6= j lim f (x) = a,

x→c+

lim f (x) = b,

x→c−

a 6= b.

The parameters i and j are chosen, as a rule, to be integers or whole numbers.

79

L

Equivalent notation 52 f is also used. See also harmonic functions and Laplace’s equation.

Lagrange’s identity Consider Liouville boundary value problem

the

[p(x)y 0 ]0 − q(x)y + λr(x)y = 0 with boundary conditions

Laplace transform A special type of integral transform, where the kernel is the function K(s, t) = Sturm- e−st . Formally, the Laplace transform of a function on the interval [0, ∞) is given by Z ∞ e−st f (t)dt. Lf (s) = 0

α1 y(0) + α2 y 0 (0) = 0, β1 y(1) + β2 y 0 (1) = 0

Laplace transform of a function can exist even if the function itself is not integrable on [0, ∞). In fact, for the existence of the Laplace transform of a function f it is enough that this function be piecewise continuous on any finite interval [0, N ] and have a growth 0 no more than exponential function: |f (t)| ≤ keat for sufficiently large values of its argument. Lagrange multiplier Assume that the function Here is a list of Laplace transforms of some common f (x, y, z) is given in some domain D and is differen- functions, where on the left is the function and on tiable there. If we want to find the extreme values of the right is its transform: this function subject to the constraint g(x, y, z) = k 1 (or, otherwise, on the level surface g(x, y, z) = k), , s>0 1 s then the gradients of both functions f and g are parallel at all extreme points, and hence, there is a real 1 number λ such that 5f = λ 5 g. The constant λ , s>a eat s − a is the Lagrange multiplier. In cases when we wish Γ(p + 1) to find the extreme values of a function subject to tp , p > −1, , s>0 two or more constraints, we will have two or more sp+1 Lagrange multipliers. For the procedure of finding a , s>0 sin at, extreme values with the use of Lagrange multipliers s2 + a2 see method of Lagrange multipliers. s cos at, , s>0 2 s + a2 Laguerre equation The second order differential a equation sinh at, , s>0 xy 00 + (1 − x)y 0 + λy = 0. s2 − a2 s In the case when λ = m, a positive integer, the solucosh at, , s>0 2 tions are polynomials, known as Laguerre polynomis − a2 als. See also Chebyshev equation. One of the most important properties of the Laplace and denote L[y] = −[p(x)y 0 ]0 + q(x)y. grange’s identity is true: Z 1 {L[u]v − uL[v]}dx = 0.

Then La-

lamina A 2 dimensional planar closed surface with transform (and some other integral transforms), is that it allows to substitute certain complicated opmass and density. erations by a simpler ones. For example, the conLaplace operator For a function f (x, y, z) of three volution of two functions is substituted by multiplivariables, which is two times differentiable by all vari- cation (see corresponding article) and the operation ables, the expression of differentiation could be substituted by the oper2 2 2 ation of multiplication by the independent variable. ∂ f ∂ f ∂ f ∆f = + 2 + 2. More precisely, let f (t) be a piecewise differentiable 2 ∂x ∂y ∂z

80 function on any interval 0 ≤ t ≤ A and satisfy the inequality |f (t)| ≤ Keat for t ≥ M , where K, a, M are some positive constants. Then the Laplace transform of f 0 exists for s > a and is given by Lf 0 (s) = sLf (s) − f (0). Under similar conditions, we have also a formula for Laplace transform of higher degree derivatives of the function: Lf (n) (s) = sn Lf (s)

use of theorems about inversions of Laplace transforms of step functions (see the next article again) gives us the final solution y=

1 [u5 (t)k(t − 5) − u10 (t)k(t − 10)]. 5

Laplace transforms of special functions (1) Let uc (t) be the unit step function (generalization of the Heaviside function) defined by

−sn−1 f (0) − · · · − sf (n−2) (0) − f (n−1) (0). These properties are crucial in applications of Laplace transform in solving initial value problems.

f (x) =

n

1 0

if if

x≥c . x

Then Laplace transform method For solving initial e−cs , s > 0. L[uc (t)] = value problems for linear non-homogeneous differens tial equations. The method is especially important in cases when the right side of the equation is discontin- In the special case c = 0(Heaviside function) the uous because the other methods (undetermined coef- transform is just 1/s. ficients, variation of parameters, series solutions) are (2) If F (s) = L[f (t)] exists for some s > a ≥ 0, then almost never successful in that case. Example: Solve L[uc (t)f (t − c)] = e−cs F (s), s > a. the equation 00 y + 4y = g(t) (3) If the function f is periodic with period T , then with the initial conditions y(0) = 0, y 0 (0) = 0 and R T −st the function g(t) defined to be 0 on the interval [0, 5) e f (t)dt and (t−5)/5 on [5, 10) and 1 on [10, ∞). This function L[f (t)] = 0 . 1 − e−sT could be written as Laplace’s equation In partial differential equations. In the simplest case of two variables, the equawhere the function uc (t) is defined below in the article tion ∂ 2 u(x, y) ∂ 2 u(x, y) Laplace transforms of special functions. Applying the + = 0. Laplace transform to both sides of the equation, using ∂x2 ∂y 2 the initial conditions, and denoting Ly = Y (s), we Solutions to this equation are called harmonic funcget −5s −10s tions. e − e . (s2 + 4)Y (s) = 2 5s latent roots The same as roots of characteristic equation or eigenvalues. Making notation K(s) = 1/s2 (s2 + 4) we will get g(t) = [u5 (t)(t − 5) − u10 (t)(t − 10)]/5,

Law of Cosines Let α, β, γ denote the (measures of) angles of some triangle ABC and denote by a (the size of) the side, opposite to α, by b– the side Now, using the partial fraction decomposition of the opposite to angle β and by c the third side (opposite function K we find its inverse Laplace transform to to γ). Then the following three relationships connect be the function k(t) = t/4 − 1/8 sin 2t. Finally, the these six quantities: Y (s) =

e−5s − e−10s K(s). 5

81 element of each row (except the rows consisting of all zeroes) is 1 and is called leading 1. leading coefficient

For a polynomial of degree n,

p(x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 the coefficient of the highest degree term, an . learning curve The same as logistic curve. See logistic differential equation. a2 = b2 + c2 − 2bc cos α, b2 = a2 + c2 − 2ac cos β, c2 = a2 + b2 − 2ab cos γ.

least common denominator Abbreviated LCD, is the least common multiple of two or more denom8 and inators of fractions. To find LCD of fractions 15 5 8 we just find the LCM of 15 and 8.

least common multiple (LCM) For given two natural numbers n and m, the least common multiple is the smallest whole number that is divisible by both of these numbers. Example, for 12 and 22, LCM(12,22)=132. There are methods of finding LCM of two or more numbers, and the most common Law of large numbers In probability and statis- is described as follows. To find LCM, find prime factics, this law states that if we chose large number of torizations of that numbers and form a number, that samples from any given distribution, then its mean is the product of all prime factors, each of whose is will be close to the mean of the distribution itself. repeated only as many times as the largest multiplicMore precisely, as the sample size increases approach- ity in any of the numbers. Example: To find LCM of ing the distribution size (or infinity, if the distribution 25, 90 and 120, we write 25 = 52 , 90 = 2·32 ·5, 120 = is continuous), then the sample mean approaches dis- 23 ·3·5 and LCM(25,90,120)=23 ·32 ·52 = 1800. There tribution mean. is a formula relating LCM with the greatest common Law of Sines Let α, β, γ denote the (measures of) factor (GCF): angles of some triangle ABC and denote by a (the n·m LCM (n, m) = . size of) the side, opposite to α, by b– the side oppoGCF (n, m) site to angle β and by c the third side (opposite to γ). Then the following relations are true: least squares regression line Suppose we have sin β sin γ sin α gathered data that came in the form of ordered pairs. = = , a b c Then each value geometrically represents a point on the plane. The problem of linear regression is to find or, equivalently, a line that represents these values the best. The meaa b c sure of ”closeness” of the line to the points from the = = . sin α sin β sin γ data is the sum of the squares of differences between actual values and corresponding values on that line. Law of Sines is used to solve oblique triangles. For Among all the possible lines the one that has the an illustration see Law of Cosines. smallest sum of that values is called the least squares leading 1’s After the process of Gaussian elimina- regression line. As any other line this line has the tion applied to a m × n matrix, the first non-zero equation of the form y = mx+b, where m is the slope Low of Cosines generalizes Pythagorean theorem, because when one of the angles is a right angle, then the corresponding cosine is zero and the formula reduces to Pythagorean. This law is used to solve oblique (not right) triangles.

82 and b is the y-intercept. Traditionally, in Statistics, this equation is written as yˆ = b0 + b1 x. The notation yˆ is used to indicate that these are ”predicted”, not the actual values from the data. The parameters b0 , b1 could be found from the data. The slope b1 = rsy /sx , where r is the correlation coefficient of the data, sy is the standard deviation of the second coordinates from the data (y-values) and sx is the standard deviation of first coordinates from the data (x-values). After the slope is found, the yintercept b0 is determined by the formula b0 = y¯−b1 x ¯, where the bar on top means the mean of the corresponding y or x-values.

is L, if for any real number ε > 0 there exists δ > 0 such that |f (x) − L| < ε as soon as |x − c| < δ, x < c. See also limit and right-hand limit. Legendre equation second order

The differential equation of

(1 − x2 )y 00 − 2xy 0 + α(α + 1)y = 0. In case α = n, a non-negative integer, the solutions of this equation are polynomials. They are called Legendre polynomials, Pn (x) if, additionally, Pn (1) = 1. Legendre polynomials Solutions of the Legendre equation for α = 0, 1, 2, ... with the initial condition Pn (1) = 1. These polynomials could be expressed in the form

least upper bound Also called supremum. If a function f or a sequence an are bounded above, then 1 dn 2 (x − 1)n . Pn (x) = n ”smallest” of all possible upper bounds is the least 2 n! dxn upper bound. Example: The sequence an = 1 − 1/n is bounded by any number M ≥ 1 but not by any number less than 1. The number 1 is the least upper legs of a right triangle Two sides of a right bound. See also greatest lower bound. triangle that form the right angle. The other side is left-hand derivative If a function is not differ- called hypotenuse. entiable at some point x = c, then the limit in the definition of derivative does not exists, i.e. lim

h→0

f (c + h) − f (c) h

does not exist. In some cases, however, one sided limit might exist. If the left-hand limit lim

h→0−

f (c + h) − f (c) h

exists, then it is called left-hand derivative of f at length of a line segment If the line segment is the point x = c. See also right-hand derivative. on the x-axis and has endpoints a and b, then the length is given by b − a. If the line segment connects left-hand limit Let f (x) be a function defined two points on the plane, then the length is just the on some interval [a, b] (also could be an open interdistance between these points. See distance formula. val ). Left-hand limit of f at some point c is the limit of f (x) as the point x approaches c from the length of parametric curve Let the curve C left, or, which is the same, as x < c. The notation be given by parametric equations x = f (t), y = is limx→c− f (x). The precise definition of left-hand g(t), a ≤ t ≤ b and f 0 and g 0 are assumed to be limit (the ε − δ definition) is the following: The func- continuous on [a, b]. Additionally, we assume that tion f has left-hand limit at the point c and that limit when t increases from a to b, the curve C is traversed

83 exactly once. Then the length of the curve C is given by the formula s Z b 2 2 dy dx + dt. L= dt dt a Example: For a circle of radius 1, we have parametric equations x = cos t,

y = sin t,

0 ≤ t ≤ 2π

and Z L=

2π

Z p sin2 t + cos2 tdt =

2π

with notation A ≤ B. If the quantities are real numbers, then this statement means that A is to the left of B on the number line, or the two points coincide. See also greater than or equal to. level curve For a function f (x, y) of two variables, the equations f (x, y) = k with any real k in the range of f , define the level curves of the function. level surface For a function f (x, y, z) of three variables, the equations f (x, y, z) = k with any real k in the range of f , define the level curves of the function. Level surfaces of functions of n variables are defined similarly.

1 dt = 2π.

L’Hospital’s rule Suppose the functions f and g are defined on some interval, containing some point length of polar curve Let the curve C be given c, with possible exception of that point. Assume also, with polar equation r = f (θ), a ≤ θ ≤ b. The the that the limit of f (x)/g(x) as x approaches c is an length of the curve is given by the formula indeterminate form 0/0 or ∞/∞. Then s 2 Z b dr f 0 (x) dθ. L= r2 + =L lim 0 x→c g (x) dθ a 0

0

Example: For the cardioid r = 1 + sin θ Z 2π p L= (1 + sin θ)2 + cos2 θdθ = 8.

implies that also lim

x→c

0

length of a space curve Suppose the curve C is given by a vector function r(t) = hf (t), g(t), h(t)i, a ≤ t ≤ b. Then the length of C is given by a formula generalizing the one for the plane: Z L=

b

|r0 (t)|dt.

a

length of a vector

This rule is the major tool of computation of limits when we have any kind of indeterminate expression. Also, if the application of derivative results in another indeterminate, then the repeated use of this rule still might produce a finite limit. Examples: (1) sin x cos x lim = lim = 1. x→0 x x→0 1 (2) 2x2 − 5 4x 2 = lim = . x→∞ 5x2 + 3 x→∞ 10x 5 lim

See norm of a vector.

less than An inequality, stating that one quantity is smaller than the other, with notation A < B. If the quantities are real numbers, then this statement means that A is to the left of B on the number line. See also greater than. less than or equal to An inequality, stating that one quantity is smaller than the other, or equal to it,

f (x) = L. g(x)

(3) x cos 2x x→0 x→0 sin 2x cos 2x − 2x sin 2x 1 = lim = . x→0 2 cos 2x 2 L’Hospital’s rule is specifically formulated for two types of indeterminate forms but can be used also lim x cot 2x = lim

84 for indeterminate forms 0 · ∞, 00 , 1∞ and others. For the first of these cases, if we need to calculate the limit limx→c f (x)g(x) and one of the functions approaches zero and the other one to infinity, then f we write the product f g as one of the quotients 1/g or 1/f g and get indeterminate form 0/0 or ∞/∞ and proceed as before. For other types the preliminary logarithming the expression results in an indeterminate form of one of the above types and after calculation of that limit we can find the original limit by exponentiating the answer. Example: To calculate the limit a x lim 1 + x→∞ x we logarithm the expression under the limit (denote it by y) we get ln y = x ln(1 + a/x) =

ln(1 + a/x) . 1/x

Then, by the l’Hospital’s rule, ln(1 + a/x) −a/x2 = lim x→∞ x→∞ (1 + a/x)(−1/x2 ) 1/x lim

a = a. x→∞ 1 + a/x

= lim

Hence, the limit in question is equal to ea . like terms In an algebraic expression with one or more variables, the terms where the variable parts are identical (constant factors may be different). For example, in the expression 2x2 y 3 − 3xy 2 + x3 y + 4xy 2 only the terms −3xy 2 and 4xy 2 are similar and could be combined. The resulting expression is 2x2 y 3 + xy 2 + x3 y. Also called similar terms. limacon, or more precisely, lima¸ con The family of parametric curves, given in polar coordinates by one of the equations r = a + b sin θ,

r = a + b cos θ.

In the spacial case |a| = |b|, we have cardioid.

limit One of the most important notions of Calculus. On the intuitive level, the limit of a function at a point or the limit of a sequence when the index grows indefinitely, could be viewed as the eventual value of the function or sequence. In the definitions that follow we will describe precisely these notions for functions and sequences. (1) Let the function f be defined on some interval I with possible exception of some point c inside of that interval. We say that f (x) has limit L as x approaches c and write lim f (x) = L,

x→c

if for any real number ε > 0 there exists another number δ > 0 such that |f (x) − L| < ε as soon as |x − c| < δ, x 6= c. (2) There are a few special cases that should be treated separately. First, if the point c is one of the endpoints of the interval I, then the limit should be substituted by one-sided limits. For definitions see left-hand limit, right-hand limit. Second, the case when the point c is the point ∞ or −∞, the definition requires modification. Here is the case c = ∞. We say that the function f (x) has a finite limit L as x approaches infinity and write lim f (x) = L,

x→∞

if for any real number ε > 0 there exists a number N > 0 such that |f (x) − L| < ε as soon as x > N . Finally, there is a case when the limit of a function

85 does not exist in a very specific way of becoming indefinitely large or indefinitely small. In these cases we say that the function has infinite limit, understanding that the limit does not exist as a finite number. One of the possible cases is the following: We say that limx→c f (x) = ∞ if for any real number M > 0 there exists a real δ > 0 such that f (x) > M as soon as |x − c| < δ, x 6= c. (3) Limits of sequences could be defined exactly as the limits of functions at infinity, if we view a sequence an as a function defined on positive integers only: an = f (n). With this view, the exact definition of the limit of a sequence is this: We say that a numeric sequence {an }, n = 1, 2, 3, ... has a limit A and write lim an = A,

n→∞

Rb integral a f (x)dx the interval endpoints a and b are called limits of integration or integration limits. line One of the basic object of Euclidean geometry along with points and planes. Lines are not defined but instead assumed to be understood as having no width and infinite length in both directions. In Cartesian coordinate system lines could be given by equations. The most general equation of the line on the plane is written in the form ax + by = c where a, b, c are any real numbers. Any line could be uniquely determined by two points (according to one of the Euclidean postulates). If the two points have coordinates (x1 , y1 ) and (x2 , y2 ) then we define the slope of the line by the relation m=

y2 − y1 . x2 − x1

This quantity does not depend on the choice of the if for any ε > 0 there exists a positive integer N such points or the order of that points. Now, the equation that |an − A| < ε as soon as n ≥ N . of the line passing through that two points could be P limit comparison test Assume that a and written using the point-slope form of the line n P bn are both series with positive terms. If y − y1 = m(x − x1 ), an =c lim or, equivalently, we could use the coordinates (x2 , y2 ). n→∞ bn In the case when the slope of the line and its interand c > 0 is a finite number, then either both series section b with the y-axis (y-intercept) are known, we converge or both diverge. can use the slope-intercept equation of the line limit laws Let f (x) and g(x) be functions such that the limits

y = mx + b.

Each of these forms could be translated into the other, hence, they are equivalent. x→c x→c The line parallel to x-axis (horizontal line) has zero exist. Then the limits of their sum, difference, prod- slope and is given by the equation y = b. The slope uct and quotient also exist and could be calculated of the vertical line (parallel to y-axis) is not defined and could be written in the form x = a. by the rules Using the notion of the slope we can see that two lim [f (x) ± g(x)] = lim f (x) ± lim g(x), lines on the plane are parallel if and only if their x→c x→c x→c slopes are equal. If two lines are given by the equations y = m1 x + b1 and y = m2 x + b2 , then they are lim f (x) · g(x) = lim f (x) · lim g(x), x→c x→c x→c perpendicular if and only if m1 · m2 = −1. limx→c f (x) f (x) In three dimensional space any plane is given by the = . lim x→c g(x) equation ax+by+cz = d and lines could be presented limx→c g(x) as intersections of two planes in the space. This apIn the last law we assume that limx→c g(x) 6= 0. proach could be extended also to lines in higher dilimit of integration In the notation of the definite mensional spaces. lim f (x)

and

lim g(x)

86 For additional information about lines see also nor- more variables where all the variables are linear. Exmal line, secant line, tangent line. ample: 2x − 3y = 5. See also systems of linear algebraic equations. line integral (1) Let C be a curve on the plane linear approximation Also called tangent line apgiven by parametric equations proximation. For a given differentiable function f at x = x(t) y = y(t) a ≤ t ≤ b some point x = a, the line that passes through the point (a, f (a)) and has the slope f 0 (a): and let ds denote the element of the length of that L(x) = f (a) + f 0 (a)(x − a). curve. Then the line integral along this plane curve could be defined as This line provides first degree approximation to the Z values of the function in small interval around the f (x, y)ds point a. C Z =

b

q f (x(t), y(t)) x2t (t) + yt2 (t)dt.

a

(2) Similarly, if C is a space curve, then we have Z f (x, y, z)ds = C

Z

b

q f (x(t), y(t, z(t))) x2t (t) + yt2 (t) + zt2 (t)dt.

a

(3) If r(t) is a vector function on the plane or in space, that defines a curve C there, then the line integral along the curve C for a vector field F is defined as Z

Z F · dr =

linear combination of vectors If v1 , v2 , · · · , vn is a set of vectors in some vector space V and c1 , c2 , · · · , cn is a set of scalars, then the expression c1 v1 + c2 v2 + · · · + cn vn is called linear combination of the set of vectors. For any system of vectors in a linear space this is another vector in V . Linear combinations of functions and matrices are defined in exactly the same way. linear dependence and independence of functions Two functions f and g are linearly dependent on some interval I if there exist two constants a and b, not both zero, such that af (x) + bg(x) = 0

b 0

F(r(t)) · r (t)dt.

for all x on I. Otherwise, the functions are linearly independent. This notion could be defined for any (4) The fundamental theorem of calculus extends to number of functions. line integrals of vector functions and has the following linear dependence and independence of vecformulation: tors A system of vectors v1 , v2 , · · · , vn is linearly Let C be a smooth curve given by the vector function dependent, if there exist scalars c1 , c2 , · · · , cn , not all r(t), a ≤ t ≤ b. If f is a differentiable function with of which are zero, such that the linear combination continuous gradient on C, then c1 v1 + c2 v2 + · · · + cn vn = 0. If there are no Z such constants, then the system of vectors is called 5f · dr = f (r(b)) − f (r(a)). linearly independent. C

a

C

linear equation (1) An equation of the form ax + b = c, where a, b, c are real (usually rational) numbers. Solution of this equation is x = (c − b)/a. Example: 2x − 3 = 6 has the solution x = (6 − (−3))/2 = 9/2. (2) An equation of two variables of the form An equation of one or ax + by = c. This equation has infinitely many

linear algebra One of the branches of the modern algebra. Deals with systems of linear equations, matrices, vectors, linear transformations, finite dimensional vector spaces and many other objects. See corresponding entries for more details. linear algebraic equation

87 etc. The equation y 000 + 2y 00 − 4y 0 + 3y = 0 is a linear homogeneous equation of the third order with constant coefficients and the equation y 00 +t2 y 0 −sin ty = cos t is a linear non-homogeneous equation of the second order with variable coefficients. The methods of solving homogeneous equations with linear inequality An expression of the form constant coefficients: ax + b ≤ c, where a, b, c are real numbers, or (1) To solve the most general second order equation any other similar expression with relations <, > or ≥ instead. Linear inequalities with two or more ay 00 + by 0 + cy = 0 (1) variables are defined exactly like the linear equations, except that the equality sign is substituted by one of we assume that the solutions should be exponential four inequality signs. functions of the form ert , where r is some unknown linear function The function, given by the constant to be determined. Substituting in the equaequation f (x) = ax + b, where a and b are any real tion gives (ar2 + br + c)ert = 0, numbers. The graph of this function is a line (hence, solutions that graphically represent a line on the plane. More generally, an equation with n variables a1 x1 + a2 x2 + · · · + an xn = b is also linear and its solution set represents an (n − 1)-dimensional hyperspace.

the term), with the slope equal to a and y-intercept b.

linear model If the mathematical model of some physical phenomena, process, etc. results in a linear equation(s) (algebraic or differential), them the model is called linear. linear operator

See linear transformation.

linear ordinary differential equations A class of ordinary differential equations where the unknown function and all of its derivatives appear in the first order. These equations could be classified further depending on the nature of coefficients (constant or variable) or the number of unknown functions and equations (single equation or system of equations), order of the equation, the presence of the ”right side” or its absence (homogeneous or non-homogeneous),

or, because ert 6= 0, we have the equation ar2 + br + c = 0 which is just an algebraic quadratic equation, called characteristic equation. Depending on the solutions of this equation, we have three possible types of solutions for the differential equation. (a) The characteristic equation has two real distinct solutions r1 and r2 . Then the functions er1 t and er2 t are the solutions of the equation (1) and the general solution of that equation is given by y(t) = c1 er1 t + c2 er2 t with arbitrary constants c1 , c2 . Example: The equation y 00 + y 0 − 6y = 0 has characteristic roots r = 2, −3 and the general solution of the equation is y = c1 e2t + c2 e−3t . (b) The characteristic equation has on real repeated root r. Then the general solution of the equation (1) is given by the formula y(t) = c1 ert + c2 tert . Example: The equation y 00 − 8y 0 + 16y = 0 has one repeated characteristic root r = 4 and the general solution is y = c1 e4t + c2 te4t . (c) The characteristic equation has two complex conjugate roots: r1 = λ+iµ, r2 = λ−iµ. In this case it is still possible to chose real solutions for the equation (1) and the general solution will have the form y(t) = c1 eλt cos µt + c2 eλt sin µt. Example: The equation y 00 +4y = 0 has complex conjugate characteristic roots r = ±2i and the general solution is y = c1 cos 2t + c2 sin 2t. (2) Solutions of equations of higher order follow the

88 same idea as the solution of the second order. To mined by the formula v = s/t, where s is the distance solve the general nth order homogeneous equation covered ant t is the time elapsed. For the special case with constant coefficients when the curve is a circle of radius r, there is a simple relationship between the linear speed v and angular an y (n) + an−1 y (n−1) + · · · + a1 y 0 + a0 y = 0 speed ω : v = rω. we form the corresponding algebraic characteristic equation an rn +an−1 rn−1 +· · ·+a1 r+a0 = 0 and find its roots. As in the case of the quadratic equations, we have only three possible outcomes described above as cases (a), (b) and (c). Accordingly, the solutions of the nth order equations are just combinations of the solutions of these three types. (3) For another approach for solving linear homogeneous equations with constant coefficients see Laplace transform method.

linear systems General term that may refer to either systems of linear algebraic equations or systems of linear differential equations. linear term In a polynomial the term that contains x. Example: In the polynomial p(x) = 2x2 − 3x + 5 the linear term is −3x.

linear transformation Let T : V → W be a function from one linear vector space to another. It is called linear transformation if for any two vectors u, v from V and any scalar c, the following conditions are Methods of solutions of non-homogeneous linear satisfied: equations with constant coefficients are described in entries (articles) undetermined coefficients,variation (a) T (u + v) = T (u) + T (v) of parameters, Laplace transform method. Arbitrary linear equations of the first order (con(b) T (cu) = cT (u). stant or variable coefficients, homogeneous or nonIn the special case when V = W , linear transformahomogeneous) could be solved by the method of intions are also called linear operators. tegrating factors. The set of all vectors in V that are transformed to Equations of order two or higher with variable coeffithe zero vector by the transformation T is called the cients are much more difficult to solve. See the entries kernel of T and denoted ker(T ). It is a subspace of Euler equation, series solutions, ordinary points, sinV in the sense that any linear combination of kernel gular points. vectors also belongs to the kernel. Also, the set of all For solutions of systems of linear equations see the vectors in W that are images of at least one vector separate entry systems of differential equations. from V is called the range of T and denoted R(T ). linear programming A branch of applied mathe- The range is a subspace of the image space W . The matics dealing with optimization problems: problems dimensions of kernel and range are called nullity and of finding the largest or smallest values of a given rank of T respectively. By one of the important thequantity. Mathematically it is reduced to finding so- orems of linear algebra, if the dimension of the space lutions of systems of linear inequalities. Among dif- V is n, then rank(T ) + nullity(T ) = n. ferent methods of solutions the most common are the A linear transformation T : V → W is called one-tographical (geometric) and simplex methods. one if it maps distinct vectors from V to distinct vectors in W . One-to-one transformations possess the linear relationship A relation between two quaninverse, denoted by T −1 and defined by the propertities (variables) where one of them depends on the ties: other linearly: If x and y are the variables then If v ∈ V , then T −1 (T (v)) = v, and y = ax + b where a and b are fixed constants. If w ∈ W , then T (T −1 (w)) = w. linear regression See least squares regression line. For additional information see also composition of linear transformations. linear speed For an object that moves along some curve. If the speed is constant then it could be deter- linearity The property of being linear. Can refer

89 The base b is a positive number, b 6= 1. The most common bases are 2, 10, and e. In the case of base 10 the logarithm is called common logarithm and in case e it is called natural logarithm. The following important properties of logarithms are true: linearization of a nonlinear equation In dif(1) logb (x · y) = logb x + logb y ferential equations, the procedure of substituting a (2) logb (x/y) = logb x − logb y nonlinear equation by a simpler linear equation. For (3) logb xr = r logb x. example, the equation of the pendulum Here x, y > 0 and r is any real number. See also 2 change of base formula for logarithms and logarithg d θ + sin θ = 0 mic function. 2 dt L to any of the situations where linear phenomenon is demonstrated. This includes algebraic equations, differential equations, transformations, and many others.

is nonlinear. To solve this equation we substitute it by the linear equation g d2 θ + θ=0 dt2 L which is much easier to solve and gives good approximation of the original problem. linearly independent solutions For differential equations. Two solutions of an equation f and g are linearly independent if from the equality αf (x) + βg(x) = 0

logarithmic differentiation Let f (x) be a positive function so that the function ln f (x) is defined. By the chain rule, f 0 (x) d ln f (x) = , dx f (x) which is the logarithmic derivative of f . This kind of differentiation is used often then the derivative of the function is more difficult to calculate. Example: If y=

x2 (x − 1)3 x+1

it follows that α = β = 0. See also linear dependence then ln |y| = 2 ln |x| + 3 ln |x − 1| − ln |x + 1| and and independence of functions. 1 dy 2 3 1 linearly independent sets Usually relates to sets = + − y dx x x−1 x+1 of vectors. See linear dependence and independence of vectors.

which results in local maximum and minimum Let the function 3 1 2 f (x) be defined on some interval that contains the 0 + − y =y point c. The value f (c) is called local maximum of x x−1 x+1 f , if there is some open interval I with center at c such that f (x) ≤ f (c) for all points x in I. Similarly, 2x(x − 1)2 (2x2 + 2x − 1) the point is local minimum, if the opposite inequality = . (x + 1)2 f (x) ≥ f (c) is true. For the method of determining local maximums and minimums see first derivative logarithmic function The function y = logb x test or second derivative test. where b > 0, b 6= 1. The domain of this function logarithm The mathematical operation that is the is (0, ∞) and the range is the set of all real numbers. inverse of the operation of exponentiation. As the ex- The inverse of the exponential function. The funcponentiation requires a base to rase to a power, loga- tion y = loge x has special notation y = ln x and in rithms also are meaningful for a base only. According case b = 10 the notation y = log x is used without to this approach, y = bx , if and only if x = logb y. indication of the base.

90 divide 462 by 3, multiplication table is difficult to apply and we use the long division. The procedure is illustrated below.

long division of polynomials When dividing a polynomial P (x) by another polynomial Q(x), where logical contrapositive If a logical statement is of the degree of Q is less than or equal to the degree of the form ”If A, then B”, (symbolically, A ⇒ B), P , the result is another polynomial plus, as a rule, a then the statement ”not B ⇒ not A”, meaning ”If B remainder. This is expressed as is not true, then A is not true” is called logical conP (x) R(x) trapositive. Direct and contrapositive statements are = D(x) + , Q(x) Q(x) equivalent in the sense that either they both are true or both are false. See also conditional statement. where D is the resulting polynomial which degree is logistic differential equation The first degree always less than the degree of P and R is the remainnon-linear differential equation of the form der polynomial which always has degree less than the degree of Q. The practical procedure of finding the y y0 = r 1 − y, polynomials D(x) and R(x) is called long division (of K polynomials) and is similar to the long division of where r and K are constants. The solution of this numbers. An example below illustrates this process. equation with initial condition y(0) = y0 is given by y= and is models logistic logistic

y0 K y0 + (K − y0 )e−rt

called logistic curve. The mathematical described by logistic equation are called models and the function growth is called In this example D(x) = x3 +2x−3 and the remainder growth. is 0. When dividing by a binomial of the form x − c the long division of numbers A procedure, algo- synthetic division is usually simpler and easier to perrithm, for dividing two numbers, usually integers form. See the corresponding article for details. or decimals. This method is used in cases when lower bound For a function f (x) a number M is simple division is difficult. For example, to divide 72 a lower bound, if f (x) ≥ M for all values of x in the by 9 we need only to remember the multiplication domain of f . table and find the answer to be 8 because we know that 8 · 9 = 72. On the other hand, if we want to lower triangular matrix A square matrix where

91

M

all the entries above the main diagonal are zeros. Example: 2 0 0 3 −1 0 . −2 5 3 MacLaurin series The special case of Taylor seThe determinant of such a matrix is just the product ries for representation of functions as a power series of all diagonal elements. See also upper diagonal ma- expanded at the point x = 0. If the function f (x) is infinitely differentiable (and also analytic) in some trix. neighborhood of the point zero, then the power series lower-upper decomposition of a matrix Or expansion is true: LU -decomposition. If a square matrix A can be re∞ X duced to row-echelon form U by Gaussian eliminaf (n) (0) n x . f (x) = tion without interchange of rows, then A can be facn! n=0 tored in the form A = LU , where L is lower triangular matrix and U is an upper triangular matrix. Most of the elementary functions have their Maclaulurking variable In Statistics. Suppose there are rin series expansions in specific intervals: two random variables related to each other in a way ∞ X 1 that the change of one of them affects the other. In = xn , (−1, 1) 1 − x n=0 other words, one of the variables is the explanatory (or independent) and the other one is the response ∞ X (or dependent) variables. In the majority of real xn x e = , (−∞, ∞) life situations, however, there are many other aspects n! n=0 (variables) that affect the change of the response vari∞ n X able. Sometimes they are disregarded by some reason n+1 x (−1) , (−1, 1) ln(1 + x) = intentionally (such as to simplify the situation) and n n=1 sometimes by omission or a mistake. These type of ∞ X variables are the lurking variable. x2n+1 (−1)n sin x = , (−∞, ∞) (2n + 1)! n=0 cos x =

∞ X

(−1)n

n=0

tan−1 x =

∞ X

x2n , (2n)!

(−1)n

n=0

(−∞, ∞)

x2n+1 , 2n + 1

(−1, 1).

See also binomial series. magnitude of a vector

See norm of a vector.

main diagonal of a matrix For a square matrix A, the imaginary line, connecting the element on the first row of the first column with the element on the nth row and nth column. Example: In the matrix 2 4 −2 A = 0 −1 1 −2 5 3

92 the main diagonal goes through the elements 2, -1 and 3.

The marginal revenue and profit functions are defined in the same way.

major axis Assume we have an ellipse with the center at the origin

mathematical induction Or the method of mathematical induction. One of the main logical tools in proving statements/theorems, especially iny2 x2 volving unlimited number of elements. It is actually + 2 =1 a2 b one of the axioms of arithmetic. with a > b. Then the foci of the ellipse are located on The method works as follows: Suppose we have a the x-axis. The line segment connecting the points a statement P (n) for each natural number n and the and −a on the x-axis is the major axis of the ellipse. following two conditions are true: In case a < b, the major axis is located along the 1. P (1) is true. y-axis. The same notion is valid also for translated 2. For every natural number k, if P (k) is true then and rotated ellipses. See also minor axis. P (k + 1) is true. Then P (n) is true for all natural numbers n. mantissa An outdated term, used to indicate the Example: Prove that for all natural numbers n, decimal number’s fractional part. Was mainly used in logarithmic and trigonometric tables to save space. n(n + 1) . 1 + 2 + 3 + ··· + n = 2 map, mapping A term used as an equivalent to the term transformation or, sometimes also function. Proof. Let P (n) be the statement we need to prove. 1(1+1) Markov chain or Markov process. A sequence of In that case P (1) will mean 1 = 2 which is true. events/processes for which the outcome in the next Now let us assume that P (k) is true: step (or stage) depends on the preceding step. It is k(k + 1) also said that in Markov chains the future states de1 + 2 + 3 + ··· + k = 2 pend only on the present state and do not depend on past. and prove that P (k + 1) is true. We have

margin of error As a rule, margin of error is defined to be twice the standard error of the sampling distribution. The interval around the mean of the distribution that has twice the length of the margin of error (one margin of error to the left and one to the right) is the confidence interval. Depending on the type of the sampling distribution (of means, of proportions, or something else) the formulas for calculating margins of errors vary significantly. For example, the standard error for proportions is dep fined as SE(ˆ p) = pˆqˆ/n and the margin of error is M E = 2SE(ˆ p). Here n is the sample size, pˆ is the proportion of ”success”, and qˆ = 1 − pˆ. marginal cost function In economics, the term marginal is used as a substitute for derivative with respect to functions under consideration. If C(x) denotes the cost function, then C 0 (x) is the marginal cost function that shows rate of change of the cost associated with producing x units of some product.

1 + 2 + 3 + · · · + k + (k + 1) = (1 + 2 + 3 + · · · + k) + (k + 1) k(k + 1) + (k + 1) 2 (k + 1)(k + 2) k(k + 1) + 2(k + 1) = = 2 2 which is exactly the statement P (k + 1). This means that P (n) is true. =

mathematical model The mathematical (algebraic, analytic, etc.) expression of real life problems or processes. Mathematical models are, as a rule, just approximations of actual processes, because in life and nature there are too many aspects to be included. The criteria of correctness and effectiveness of a mathematical model is its accuracy. If the model results in solutions that (more or less) accurately describe the natural phenomenon then it

93 is considered a good model. Depending on nature of the problem, the model will result in algebraic, trigonometric, differential, or some other type of equations. matrix Plural: matrices. Formally, a matrix is a collection of numbers, vectors, functions, etc., organized in m rows and n columns and separated by a pair of parentheses, braces, or other convenient separation symbols. A numeric m × n matrix has the form a11 a12 . . . a1n a21 a22 . . . a2n . .. .. .. . . am1

am2

...

amn

where the coefficients {ajk } are real or complex numbers. The following operations are defined for matrices (more details and examples are provided in corresponding special entries, such as addition and subtraction of matrices, multiplication of matrices, adjoint of a matrix, inverse of a matrix, and many others): (1) The equality of two matrices A and B means that they have the same size (the number of rows and columns are the same for them) and that all elements in the first matrix are equal to the corresponding elements in the other matrix. (2) Addition (subtraction) of two matrices is possible only if they have the same size. In that case we just add (subtract) elements at identical positions. (3) Multiplication of a matrix by a scalar is defined as multiplication of each element of the matrix by that scalar (constant, number). Division by a scalar is defined similarly. (4) Multiplication of the matrix A (from the left) by the matrix B (from the right) is possible only if the number of columns in A is equal to the number of rows in B. Hence, if A is of size m × k and B is of size k × n, then the resulting matrix will have the size m × n. The element of the product in the ith row and jth column (denoted by cij ) is equal to ai1 b1j + ai2 bj2 + · · · + aik bkj . Generally speaking, multiplication of matrices is not commutative. See examples in the entry multiplication of matrices. (5) For square matrices (the same number of rows

and columns) the main diagonal consists of elements located on positions with the same numeric value for rows and columns: ajj , j = 1, 2, · · · , n. The matrix that has all 1’s on the main diagonal and zeros elsewhere is the identity matrix of size n. The inverse to a matrix A is a matrix B which multiplied by A (from both left and right) results in identity matrix. If the inverse exists, then it is unique and is denoted by A−1 . If the inverse exists, then the matrix is called invertible, otherwise it is called singular or noninvertible. (6) The transpose of a matrix is a matrix for which the roles of the rows and columns are interchanged. If A is of size m×n, then its transpose is of size n×m. For additional properties and operations with matrices see also augmented matrix, Gaussian elimination, Gauss-Jordan elimination, adjoint of a matrix, eigenvalues, eigenvectors, similar matrices, zero matrix. matrix function A matrix where each entry is a function. The matrix cos θ − sin θ A= sinθ cos θ describes rotations of the plane and is an example of matrix function. matrix method For solving systems of linear algebraic equations. If the system has equal number of equations and unknowns (square system), then it could be written in the matrix form Ax = b, where A is a square matrix of size n×n and x, b are column vectors of size n. Now, if A is invertible, then applying its inverse A−1 from the left to this equation we will get x = A−1 b and this is the matrix solution of the system of equations. matrix transformation A linear transformation T : V → W of one vector space to another that could be written as T v = M v, where M is some matrix. If V and W are the Euclidean spaces Rn and Rm respectively, then any linear transformation is a matrix transformation. maximization problems The problem of finding the largest value of some variable quantity. See also

94 {2, 7, 3, 9, 6, 2, 4} we first put them in order, repeating the numbers that appear more than once: 2, 2, 3, maximum and minimum values The largest or 4, 6, 7, 9. Now, the number 4 (which belongs to the smallest values of a function on a given interval. Deset S) is the median, because we have three numbers pending on type of the interval (closed or open) and less than 4 and three numbers greater than 4. the function (continuous or not), the function may (2) For the set S = {2, 7, 3, 5, 9, 6, 2, 4} we order it or may not take that values. Additionally, we distinand get 2, 2, 3, 4, 5, 6, 7, 9. Now, because the numguish the absolute maximum and minimum and local ber of elements is even, there is no number in the set maximum and minimum values. For the methods of with the property of being in the middle. In this case finding these values see corresponding entries. we take two ”middle” ones 4 and 5 and their arithmean Average of a set of objects (numbers, func- metic mean (4+5)/2=4.5 will be the median of S. In tions, etc.). See arithmetic mean and geometric mean this case median does not belong to S. for definitions. method of cylindrical shells One of the methods mean value theorem Let f (x) be a continuous of evaluating the volumes of three-dimensional solids function on some closed interval [a, b] and assume it that are solids of revolution. Suppose we need to find is differentiable on open interval (a, b). Then there the volume of a solid that is formed by rotating the exists (at least one) point c, a < c < b, such that curve y = f (x), a ≤ x ≤ b, about the axis x = c. The method consists in considering small portions of f (b) − f (a) 0 the interval [a, b] with the height f (x) with x being . f (c) = b−a some point in that portion, rotated about the same The geometric meaning of this theorem is that there axis x = c. The result is a cylindrical shell. If we should be at lest one point on the graph of the func- calculate the volumes of that shells and add all the tion where the tangent has the same slope as the line small shells’ volumes, the result will be the approximate volume of the solid. In the limit, as the sizes segment connecting two endpoints. of the bases become smaller and the number of shells mean value theorem for integrals If the func- increase, we will get the exact volume of the solid tion f (x) is continuous on some interval [a, b], then which is now given by the integral there exists a point c, a < c < b, such that Z b Z b V = 2π r(x)f (x)dx, f (x) dx = f (c)(b − a). a optimization problems.

a

where r(x) is the distance from the point x to the axis of rotation. Example: Consider the region bounded by the lines y = x/2, y = 0, x = 2. Find the volume of the solid that is obtained by rotating this region about the axis x = 3. Solution: The region is a triangle with vertices at the origin and points (2, 0), (2, 1). If we now use the method of cylindrical shells, we will have r(x) = 3−x median For a set of numeric values, the value that and Z 2 is in the middle of that set in the sense that there x V = 2π (3 − x) dx are exactly equal numbers in the set that are greater 2 0 than median or less than the median. The median may or may not be an element of the set. Examples: x3 2 64π = 3x2 − |0 = . (1) To find the median for the set of numbers S = 6 3 measure Measure on a set is a way (a rule) to assign a positive number to each subset of that set (including the set itself). Intuitively, a measure of a set on the real line is its ”length”, the measure of a plane set is its area, and the measure of a three-dimensional solid is its volume. Exact definition of measure of a set is given in advanced analysis courses.

95 method of Lagrange multipliers Allows to find the extreme values of a function of several variables using the so-called Lagrange multipliers: Theorem. To find the maximum and minimum values of the function f (x, y, z) subject to the constraint g(x, y, z) = k (assuming that these extreme values exist on that surface and 5g 6= 0): (1) Find all the values of x, y, z, λ such that

with a > b. Then the foci of the ellipse are located on the x-axis. The line segment connecting the points b and −b on the y-axis is the minor axis of the ellipse. In case a < b, the minor axis is located along the x-axis. The same notion is valid also for translated and rotated ellipses. See also major axis.

minors of matrix Let A be a square matrix with entries {aij }. The minor of matrix A corresponding 5f (x, y, z) = λ 5 g(x, y, z) to the element aij is the determinant of the matrix that remains after we erase the ith row and jth coland umn of the matrix A. Example: If g(x, y, z) = k (2) Evaluate the function f at all points (x, y, z) that −4 4 1 are found in the previous step. The largest of these A = 0 −1 3 values is the maximum value of f and the smallest is 2 5 2 the minimum value of f . then the minor corresponding to the element a23 method of least squares See least squares regres- (which is 3) is sion line. −4 4 midpoint formula If (a, b) and (c, d) are two det = −20 − 8 = −28. 2 5 points on the plane, then the point on the middle of line segment connecting these points has the coor- See also cofactor. b+d . , dinates a+c 2 2 mixing problems Or mixture problems. A genmidpoint rule For approximate evaluation of an eral name for a category of application (word) probintegral. It the function f (x) is defined on some fi- lems where the objective is to find the amount of two nite interval [a, b], then we divide that interval into n ingredients to be mixed to make a mixture with specequal parts of the length ∆x = (b − a)/n, as in the ified properties. Examples: (1) What quantity of a definition of the Riemann integral. Then denote the 60% acid solution must be mixed with a 30% solution midpoint of the interval [xi , xi+1 ] by xi , 1 ≤ i ≤ n. to get 6 liters of 50% solution? Solution: Denote by x With this notations the midpoint rule states that the amount of the 60% solution and by y the amount Z b n X of the 30% solution. Then the amount of the acid in f (xi ). f (x)dx ≈ ∆x x liters will be 0.6x liters (because only 60% of it is a i=1 actually acid) and the amount of acid in the y liters This method is less accurate than the trapezoidal rule will be 0.3y liters. In the 6 liter mixture the amount or the Simpson’s rule. The midpoint rule is easily of the acid will be 0.5 × 6 = 3 liters. This results in extended to functions of more than one variables in the equation 0.6x + 0.3y = 3 and the second relation a very similar way. connecting the two variables is x + y = 6 because the minimization problems The problem of finding total amount is 6 liters. This simple system of linear the smallest value of some variable quantity. See also equations has the solution x = 4, y = 2. (2) A jeweler has a ring weighing 90 g made of an optimization problems . alloy of 10% silver and 90% of gold. He wants to minor axis Assume we have an ellipse with the use this ring to make another piece of jewelry with center at the origin gold content of 75%. How much silver he needs to 2 2 add? Solution: Denote by x the amount of silver to y x + = 1 be added. Then the weight of the new alloy will be a2 b2

96 90 + x grams and the amount of gold there will be 0.75(90 + x) grams. On the other had this new mixture still contains the same amount of gold as before, which is 90% of 90 grams, equal to 81 grams. Hence, the equation will be 0.75(90 + x) = 81. The solution is x = 18 grams. mode For a set of numeric values, the number that appears most frequently. A set may have multiple modes or have none at all. Examples: The set S = {1, 2, 3, 4, 2, 5, 2, 6} has one mode which is 2. The set S = {1, 2, 3, 4, 3, 5, 2, 6} has two modes: 2 and 3, and the set S = {1, 2, 3, 4, 5, 6} has no mode, because no numbers appear more than once. In this last case it also could be said that all the elements of the set are modes. model, mathematical See mathematical model.

variables. If a function f depends on more than one variable, then the generalization of Riemann sums and Riemann integrals allows to integrate this function in a domain in n-dimensional Euclidean space Rn . The most common cases are the double integral and triple integral. See corresponding entries for details of definitions. multiplication of complex numbers (1) Let z = x + iy and w = u + iv be two complex numbers written in the standard form. Then, to multiply these two numbers we multiply all terms with each other and combine the real and imaginary parts: z · w = (x + iy)(u + iv) = (xu − yv) + i(xv + yu), because i2 = −1. Example: (3 − 2i)(−1 + 4i) = −3 + 2i + 12i − 8i2

= −3 + 8 + (2 + 12)i = 5 + 14i. modulus of a complex number For a complex number z = x + iy, the non-negative number |z| = (2) If the complex numbers z = r(cos θ + i sin θ) and p x2 + y 2 . Shows the distance from the point z on w = ρ(cos φ + sin φ) are given in trigonometric form, the complex plane to the origin. Also sometimes then called absolute value of the complex number. z w˙ = rρ[cos(θ − φ) + i sin(θ − φ)]. monomial An algebraic expression that contains multiplication of fractions The product of two constants multiplied by natural power(s) of one or fractions is defined to be another fraction with the more variables. Examples: 2x2 , −3x3 y 2 , xyz. numerator being the product of two numerators and the denominator the product of denominators. Formonotonic function A function that is either inmally, if ab and dc are the given fractions, then creasing or decreasing. See corresponding entries. a·c a c monotonic sequence A sequence that is either in· = . b d b·d creasing or decreasing. See corresponding entries. monotonic sequence theorem If a monotonic sequence is bounded, then it has a finite limit.

In practice, instead of multiplying directly, we simplify first and then multiply, to avoid possible big numbers. Example:

multiple-angle formulas In trigonometry, formu12 15 12 · 15 3·3 9 las relating trigonometric functions of multiple angles · = = = . 25 32 25 · 32 5·8 40 with functions of single angle. The most common are the double angle formulas, but triple or quadruple multiplication of functions For two functions angle formulas also exist. Here is one of possible ver- f (x) and g(x) their product (result of multiplication) sions of the triple angle formula for cosine function: function is defined to be the product of their values: (f g)(x) = f (x)g(x). This function is defined where cos 3θ = 4 cos3 θ − 3 cos θ. f and g are both defined. Other versions of this formula and formulas for other multiplication of matrices Let A be an m × r functions also exist but very rarely used. matrix and B an r × n matrix. Then the product multiple integral For functions of several real AB = A · B is an m × n matrix whose entries are

97 determined by the following rule: The entry in row i and column j is the sum of products of all pairs from ith row and jth column. In more detail. Let A = {aij } and B = {bij }.P Then the product matrix r C = AB has entries cij = k=1 aik bkj . As seen from the definition, matrix product is possible only when the number of columns in the first matrix is equal to the number of rows in the second one. In particular, matrix multiplication is not commutative, because in most cases BA is not even defined when AB is defined. For square matrices A, B both products AB and BA are defined but are not equal in general. Examples: (1)Let 4 1 4 3 1 2 4 A= , B = 0 −1 3 1 . 2 6 0 2 7 5 2 Then

AB =

12 27 8 −4

30 13 26 12

2 1

−3 4

−6 8

16 1

A=

,

0 2

B=

.

multiplication property of equality Let A, B, C be any algebraic expressions and assume also that C 6= 0. Then, if A = B, then A · C = B · C.

5 −1

then AB =

,

BA =

x3 x4 x4 x2 − x2 − x3 + + x4 − + + ··· 2 6 2 24 x2 5x3 7x4 =1+x− − + − ···. 2 6 24

=1+x+

(2) If

multiplication of power series Assume the functions f (x) and g(x) are represented by power series. Then the product of these functions could be represented as a power series (with the interval of convergence being equal to intersection of two intervals of convergence) and that series is the product of the power series of the functions f and g. Example: To multiply the functions ex and 1/(1 + x2 ) we write their power series representations and multiply as we would multiply two polynomials: x2 x + · · · 1 − x2 + x4 − x6 + · · · 1+ + 1! 2!

5 3

20 −10

and hence, AB 6= BA.

multiplication property for inequalities Let A, B, C be any algebraic expressions and assume also that C 6= 0. (1) If A ≤ B and C > 0, then A · C ≤ B · C; (2) If A ≤ B and C < 0, then A · C ≥ B · C. Similar statements are true also in the case of other types of inequalities: <, >, ≥. Example: 2 < 5 but 2 · (−3) = −6 > 5 · (−3) = −15.

multiplication of polynomials To multiply two polynomials we just multiply all terms of the first one by all terms of the second one and combine all like multiplication rule for probabilities Let A and B be two independent events. Then the probability terms. Formally, if that both of these events will happen is given by the n m X X formula p(x) = ai xi , q(x) = bj xj , P (A ∩ B) = P (A) · P (B). i=0

j=0

then the product is a polynomial of degree n + m given by n+m X p(x) · q(x) = ck xk , k=0

where ck =

Pk

i=0

ai bk−i . Example:

(x2 + x − 2)(x3 − x + 1) = x5 + x4 − 3x3 + 3x − 2.

This formula extends to any number of independent events. For the case when the events are not independent the general multiplication rule for probabilities works. Another notation for multiplication rule is P (A andB) = P (A) · P (B). multiplicative identity An element, that multiplied by any other element of the given set, does not change it. For the sets of real or complex numbers

98 the real number 1 serves as a multiplicative identity. For the set of square matrices, the identity matrix is the multiplicative identity. multiplicative inverse An element b of some set is the multiplicative inverse of some other element, a, if a · b = b · a = 1, where by 1 the multiplicative identity is denoted. For the set of real numbers the inverse of a is denoted by a−1 or 1/a and it exists for all numbers except a = 0. For matrices, the inverse is denoted by A−1 .

N natural exponential function The exponential function with the base equal to e and notation y = ex . In calculus, natural exponential functions allow to x 0 x simplify R many calculations. In particular, (e ) = e and ex dx = ex + C.

natural growth law See exponential decay, expomultiplicity of eigenvalue See eigenvalues of nential growth. matrix. natural logarithmic function The logarithmic function with the base equal to e and special notamultiplicity of a zero Let p(x) be a polynomial of tion: log e x = ln x. In calculus, natural logarithms degree n ≥ 1. Then, by the Fundamental theorem of allow to simplify many calculations. In particular, Algebra it could be factored as a product of exactly n we have (ln x)0 = 1/x and linear binomials: p(x) = a(x − c )(x − c ) · · · (x − c ), 1

2

n

where c1 , c2 , ..., cn are some complex numbers and Z p(ck ) = 0, k = 1, 2, ...n. If any of these factors repeat, ln x dx = x ln x − x + C. then the corresponding cj is called multiple zero of the polynomial. The number of times this factor appears in factorization is the multiplicity of this zero. natural number The numbers 1, 2, 3, 4 · · ·. This Example: In polynomial p(x) = (x−1)(x+1)2 (x−2)3 sequence of numbers goes indefinitely and there is no the zeros are x = 1, x = −1, x = 2 and they have greatest number in it. Also are called counting nummultiplicities 1,2 and 3 respectively. bers. multiplying factor See integrating factor. negative angle If the angle is placed in standard mutually exclusive events Two events are mu- position and the terminal side is moved in negative tually exclusive, if they cannot happen at the same (clockwise) direction, then the angle is considered time. Complementary events are always mutually negative. See also angle. exclusive, but this is also true for other events. negative definite function A function F (x, y) When rolling a die, the outcomes 2 and 4 are not defined on some domain D containing the origin is complementary but they are mutually exclusive. negative definite, if F (0, 0) = 0 and F (x, y) < 0 everywhere else in D. If F (x, y) ≤ 0, then the function is negative semidefinite. See also positive definite. negative definite matrix A symmetric matrix A is negative definite, if for any vector x 6= 0, xT Ax < 0. If the inequality is substituted by ≤, then the matrix is negative semidefinite. Here xT indicates the transpose of the column vector x. See also positive definite matrix. negative number A number that is less than zero. Equivalently, any number that could be placed on the number line to the left of the origin.

99 Newton’s law of cooling According to this law, discovered experimentally, the rate of cooling (loss of temperature) of a body is proportional to the difference of temperatures of the body and the surrounding environment (if this difference is relatively small). Mathematically, this law could be expressed by the differential equation dT (t) = −k(T − T0 ), dt

their signs, |f 0 (x)| ≥ m > 0 and |f 00 (x)| ≤ M for all values of x in some interval (x0 , x1 ). Assume that f (x0 ) and f (x1 ) have opposite signs, while f (x1 ) and f 00 (x1 ) have the same sign. Then there exists a point c in (x0 , x1 ) with f (c) = 0. Moreover, if |x1 − x0 | ≤ m/M , then the successive approximations by Newton’s method x1 , x2 , x3 , · · · approach c and |xn+1 − c| ≤ |xn+1 − xn |.

Newton’s second law The force on the object is equal to its mass multiplied by the acceleration of where T (t) is the temperature of the body and T0 is that object. the surrounding temperature. The solution is given nilpotent matrix A square matrix A such that by T (t) = T0 + (T (0) − T0 )e−rt . An = 0 for some positive integer n ≥ 1. Newton’s laws of motion The laws of classical nondegenerate conic section The opposite of mechanics established by I.Newton. See also Newdegenerate conic sections. All the regular conic secton’s second law. tions (ellipses, parabolas, hyperbolas) are nondegenNewton’s method A method for finding approx- erate. imate solutions of some equation f (x) = 0, where f nondifferentiable function A function that is is some sufficiently smooth function, not necessarily not differentiable. Among the reasons for a function a polynomial. If the solutions of this equation are to be nondifferentiable are: failure to be continuous not possible by some exact method, then in many at the point, having ”wedges” at the point. There cases Newton’s method gives good approximations. are continuous functions, that are not differentiable The procedure is as follows. If we are trying to find at any point. the solution of the equation and we have two values a and b, where f (a) and f (b) have opposite signs, nonhomogeneous algebraic equations The syswe know (by the intermediate value theorem) that tem of linear equations where not all right sides are there is a solution to the equation f (x) = 0 in the zero. See also homogeneous algebraic equations. interval (a, b). Choose any of the endpoints to be our nonhomogeneous linear differential equation first approximation to the solution and denote it by The equation of the form x1 . If f (x1 ) 6= 0, then we choose the second point of approximation using the intersection of the tangent y 00 + p(x)y 0 + q(x)y = g(x), line to f at the point x1 with the x-axis. This point, which we denote by x2 , will be given by equation or a similar one with higher order derivatives. In case x2 = x1 − f (x1 )/f 0 (x1 ). Continuing the same way, when g(x) = 0, the equation is homogeneous. we have a sequence, given by recurrence formula noninvertible matrix Also called singular matrix. f (xn ) A matrix such that the inverse does not exist. See , n = 1, 2, 3, ... xn+1 = xn − 0 f (xn ) also corresponding entry. This procedure does not always give good approximation to the solution. The following statement indicates conditions under which the procedure converges to the solution and also gives estimate of how exact the approximation is. Theorem. Suppose f 0 (x) and f 00 (x) do not change

nonlinear differential equation Any differential equation where the unknown function or any of its derivatives appear in a nonlinear form. Examples could be y 00 + 2(y 0 )2 = 0,

y 00 + y 0 y = 2.

100 The nonlinear equations could be categorized by the highest degree of the derivative involved, by the existence or non-existence of the function on the right side, and many other criteria, just as in the case of linear equations. The theory of nonlinear equations is not as complete as the theory of linear equations and solving different equations requires involvement of different methods. These include the method of linearization, and solutions of autonomous equations and systems. For more details see the corresponding entries.

normal distribution The continuous probability distribution expressed by the normal density function. The probabilities for this distribution are calculated as areas under the curve. For example, if we want to know what is the probability of choosing a point between the values a and b, then we calculate the area under the curve between a and b. These areas are calculated with the use of tables, graphing calculators or computers. See also standard normal distribution.

nonsingulal matrix A (square) matrix, that has an inverse. Opposite of the singular matrix. nontrivial solution Many differential equations have two types of solutions: one is a meaningful function and the other one is a constant or even identically zero solution. These last type solutions are called trivial and the first type solutions are nontrivial solutions. Example: The equation y 0 = −y 2 has the trivial solution y = 0 and nontrivial solutions y = 1/(x + C) for any real constant C. norm of a vector If v is a vector in some vector space V with coordinates (v1 , v2 , · · · , vn ), then its norm is defined to be the non-negative number q ||v|| = v12 + v22 + · · · + vn2 .

normal matrix A matrix A is normal, if AA∗ = A∗ A. Here A∗ denotes the conjugate of the transpose of the matrix. In case of real matrices, A∗ is just the transpose of A.

normal probability curve The curve that is the graph of the normal density function. See the definiIn the case when V is an inner product vector space, tion and the picture above. ||v|| = hv, vi1/2 . Also called the length or magnitude of the vector. normal system For any system of linear equations written in the matrix form Ax = b the associated normal The same as normal vector. equation normal density function The function that deAT Ax = AT b scribes the normal distribution. Mathematically it is is called the normal system. Here AT is the transpose given by the function of the form of A. The normal system is important in finding the (x−a)2 least square solutions of the system. f (x) = ce , normal vector A vector that is orthogonal to a where c and a are constants specifically chosen to curve or a surface. In case of the plane curve this satisfy the conditions of probability distributions. For means that the vector is perpendicular to the tanmore precise definition see bell shaped curve. gent line at a given point of the curve. In three dimensional space orthogonality neans that the vector normal derivative The directional derivative in is perpendicular to the tangent plane to the surface the direction of the normal vector. at some point of that surface. A line in the direction

101 of the normal vector is called normal line.

number theory The branch of mathematics that deals with the properties of whole numbers. Despite the fact that this is one of the oldest branches of mathematics, there are several unresolved problems, concerning properties of whole numbers. Among them, the famous Riemann hypothesis deals with properties of prime numbers.

normalization To make a system of orthogonal vectors orthonormal, the procedure of normalization is necessary. For that we just divide each vector from orthogonal system by its norm and the resulting vectors will have norm 1. See Gram-Schmidt process for the details. This term can also be used in other contexts where the notion of ”normal” is appropriately numerator For a fraction or a rational function, defined. the top part in the fractional expression. In number 7 , 7 is the numerator and 13 is called denominator. nullity Let T be a linear transformation in some 13 In function vector space V and K denotes the kernel of that 3x4 − 2x2 + x − 5 transformation. The dimension of the kernel is called x5 + 2x2 − 1 nullity. the polynomial 3x4 − 2x2 + x − 5 is the numerator. null hypothesis In statistics, a hypothesis (statement, assumption) that a certain population param- numerical coefficient Same as coefficient. eter has some value. This hypothesis is usually de- numerical integration Calculation of definite noted by H0 and has the form parameter=some value. integrals approximately, when exact calculations are For example, if the hypothesis is about the population impossible or difficult. The same as approximate mean, then H0 : µ = µ0 . Sometimes the hypothesis integration. is given also in the form µ ≥ µ0 or µ ≤ µ0 . See also numerical methods General term for indicating alternative hypothesis, hypothesis testing. various methods of solving equations when analytic null set The set that contains no elements. Has the (sometimes algebraic) solutions are either impossible same meaning as empty set and the same notation ∅. or very complicated. For solutions of algebraic nullspace Let A be an m × n matrix and x be a equations see, e.g., Newton’s method. For numerical vector in Rn . The solution space of the equation solutions of differential equations see Euler method. Ax = 0 is the nullspace of the matrix. The nullspace is the subspace of Rn which is translated to zero by the action of the matrix (linear transformation) A. See also kernel of linear transformation. number The most basic object in mathematics. The numbers are impossible to define, they rather could be described. We distinguish two major sets of numbers: the real numbers and the complex numbers with the understanding that the first set is a subset of the second one. Additionally, the real numbers consist of subsets of irrational and rational numbers. Rational numbers further have the subsets of integers, whole numbers, natural numbers and prime numbers. See the corresponding entries for definitions. The visual presentation of real numbers is done by expressing them as points on the real line, and the complex numbers can be expressed as points on the complex plane.

102

O oblique asymptote Also called slant asymptote. The line y = ax + b is an oblique asymptote for the function f (x) if one or both of the relations limx→∞ [f (x) − (ax + b)] = 0 or limx→−∞ [f (x) − (ax + b)] = 0 hold. This relations mean that for large values of x (positive or negative) the value of the function is approximately the same as the value of the linear function y = ax + b. A function may or may not cross its oblique asymptote. Examples: (1) The function 2 2 +1 = (x − 1) + x+1 has oblique asymptote f (x) = xx+1 y = x − 1 but it never crosses that line. (2) The sin x function f (x) = x − 1 + 1+x 2 has the same oblique asymptote y = x − 1 and they intersect infinitely many times. oblique triangle A triangle where no two angles and no two sides are equal observational study In statistics, when the researcher collects data by just observing the objects, without actively trying to affect the outcome. Surveys and polls are examples of observational odd permutations A permutation that is the result of odd number of transpositions. Equivalently, studies. could be defined as a result of odd number of inverobtuse angle An angle that measures between sions. 90◦ and 180◦ in degree measure or between π/2 and one-sided limit See left-hand limit and right-hand π in radian measure. limit. obtuse triangle A triangle that contains an one-step methods In approximate (numeric) soobtuse angle. lutions of differential equations any method that reoctant The Cartesian coordinate system in the quires the knowledge of values only in one previous tree-dimensional space creates tree coordinate planes step to find values on the next step. The most comgiven by the equations x = 0, y = 0 and z = 0 which mon example is the Euler method. intersect at the origin. These three planes divide the one-to-one function Let f (x) be a function despace into eight parts called octants. fined on some (possibly infinite) interval [a, b]. It odd function A function f (x) of real variable is one-to-one on the interval, if for any two points that satisfies the condition f (−x) = −f (x). This x1 6= x2 inside that interval f (x1 ) 6= f (x2 ) and if condition means that the graph of the function is f (x) = f (y), then x = y. In other words, one-to-one symmetric with respect to the origin. The functions functions cannot take the same value more than once. f (x) = sin x and f (x) = x3 are examples of odd Examples: The functions f (x) = ax + b, f (x) = functions. 2x , f (x) = ln x are all one-to-one. On the other

103 hand, the function f (x) = x2 is not one-to-one, be- the order of integration. 2 cause for any√positive number √ a, x = a gives the so- order of operations Set of rules that determine lutions x = a and x = − a. One-to-one functions the way arithmetic and algebraic operations are perare important because they possess inverse functions. formed to avoid ambiguity in calculations. According one-to-one transformation A linear transforma- to this rules the operations are done in the following tion T : V → W is one-to-one, if it transforms differ- succession: ent points of V to different points of W . One-to-one 1) Parentheses (includes also brackets and braces) transformations are invertible. from inside out; 2) Exponents; open interval Intervals of the form (a, b) where 3) Multiplication and division (left to right); the endpoints are not included. In this case one 4) Addition and subtraction (left to right). or both endpoints may be infinite, so the intervals Example: 25 − 23 · 3 ÷ (8 + 4) = 25 − 23 · 3 ÷ 12 = (a, ∞), (−∞, b) and (−∞, ∞) are considered open. 25 − 8 · 3 ÷ 12 = 25 − 24 ÷ 12 = 25 − 2 = 23. open region A region is called open if for any point ordered pair Two real numbers a and b grouped P in that region there exists a circle with center at together and enclosed in parentheses: (a, b). The orthat point that lies completely in the region. The der of the numbers is important and (a, b) is differ2 2 region D = {x + y < 1} is open but the region ent from (b, a). Each ordered pair defines uniquely 2 2 G = {x + y ≤ 1} is not, because for the latter for a point in the Cartesian plane and, conversely, each 2 2 points P on the boundary x + y = 1 any circle with point corresponds to an ordered pair where the numthat center is only partially contained in the region. bers are the coordinates of the point. This notion operator See linear transformation. extends to ordered triples also where they now correspond to points in three dimensional space. Simiopposite A term that could be used with different larly, we can generalize it to the case of ordered nmeanings in different situations. Most commonly is tuples which correspond to points in n-dimensional used as an opposite of a real number a which is −a. Euclidean space. The opposite of 5 is −5 and the opposite of −2 is −(−2) = 2. ordinary differential equation A differential equation that involves only functions and derivatives optimization problem Any problem where the of functions of one independent variable. objective is to find the greatest or smallest possible values of certain variable quantity. ordinary point For linear differential equations with variable coefficients. Suppose we need to find order of a polynomial See degree of a polynothe series solution for the equation mial. P (x)y 00 + Q(x)y 0 + R(x)y = 0 order of differential equation The highest order of derivative present in the given differential equaabout the point x = x0 . That point is called ordinary tion. In the equation point if P (x0 ) 6= 0. See the entry series solution for 000 2 0 details how this fact is used in finding solutions. y − 2x y + sin xy = cos x the order is 3 because y 000 is the highest order derivative in that equation.

ordinate In the plane Cartesian coordinate system, the name of the y-axis.

orientation of a curve See positive orientation of order of integration In double (or, more genera closed curve. ally, multiple) integrals the order in which the repeated (iterated) integration is performed. See Fu- orientation of a surface See positive orientation bini’s theorem for conditions justifying the change of of a closed surface.

104 origin In any coordinate system, the point from where we start measurements. In particular, in Cartesian system origin is the point usually denoted by number 0. In the case of the plane system, for example, the origin is the point of intersection of x− and y−axes and has the coordinates x = 0 and y = 0. Hence, the origin is the point corresponding to the ordered pair (0, 0).

matrix A form an orthonormal set in Rn . Example: The matrix √ 3/2 √ −1/2 3/2 1/2 is orthogonal.

orthogonal projection Let v be a vector in Rn and suppose we map that vector onto a line formed orthogonal A term that substitutes the term perby another vector a not parallel to v. Denote the pendicular when we deal mainly (but not exclusively) resulting vector by proja v = b. If this mapping is with vectors. performed in such a way that the vectors v and b orthogonal basis A basis {u1 , u2 , . . . , un } of a are orthogonal, then the mapping is called orthogonal linear space V is said to be orthogonal if any pair projection. The orthogonal projection of a vector on of vectors in that basis is orthogonal, i.e. ui · uj = 0 a plane is defined in a similar manner. for any 1 ≤ i, j ≤ n, i 6= j. See also basis. Example: The operator of orthogonal projection of any vector (x, y) on the plane onto the x-axis maps orthogonal complements Let V be an inner that point onto the point (x, 0). The standard matrix product vector space and W its subspace. The set of that operator is of all the vectors in V that are orthogonal to W is the orthogonal complement of W . The orthogonal 1 0 complement of any subspace is itself a subspace. 0 0 orthogonal curves Two curves are orthogonal at a point where they intersect, if the tangent lines of orthogonal vectors Two vectors u and v are orthat curves at that point are perpendicular. Examthogonal if they form a 90◦ (π/2) angle. Equivalently, ple: The hyperbola y = 1/x and the line y = x are cosine of the angle between these two vectors is zero. orthogonal at their poit of intersection (1, 1). This last condition is also expressed with the use of orthogonal diagonalization Let A be a square the dot product: matrix and assume that there exists a orthogonal mau·v = 0, cos θ = trix P such that the matrix P −1 AP = P T AP is di||u||||v|| agonal. Then A is called orthogonally diagonalizable and the process is called orthgonal diagonalization. where ||u|| is the norm (magnitude, length) of the It is known that being orthogonally diagonalizable is vector u. equivalent of being symmetric. orthogonality of functions Two functions f (x) orthogonal matrices A square n × n matrix A and g(x) defined on some interval [a, b] (finite or inis orthogonal if its inverse is equal to its transpose: finite) are called orthogonal, if A−1 = AT . Equivalently, AAT = AT A = I, the Z b identity matrix. Orthogonal matrices have many imf (x)g(x)dx = 0. portant properties listed below: a (1) The inverse of an orthogonal matrix is orthogoMany systems of functions are orthogonal in this nal; Among them the trigonometric system (2) The product of orthogonal matrices is orthogonal; sense. (3) The determinant of an orthogonal matrix is equal {cos nx, sin nx}, n = 0, ±1, ±2 · · · (on the interval [0, 2π]), Chebyshev polynomials (on the interval 1 or -1; (4) Both the row vectors and column vectors of the [−1, 1]), Bessel functions, and Legendre polynomials

105 (on appropriate intervals).

P

orthonormal basis A basis {u1 , u2 , · · · , un } of a linear space V is said to be orthonormal if any pair of vectors in that basis is orthogonal (in other words, the basis is orthogonal), and each of them has norm equal to one: ||uj || = 1, j = 1, 2, · · · , n. See also basis paired data In statistics has the same meaning as and Gram-Schmidt process. the ordered pairs in algebra. The distinction is made osculating plane For some smooth space curve let because in statistics the paired data comes from colN and T denote the normal and tangent vectors at lection of samples while in algebra the ordered pairs some point respectively. Then the plane formed by need not have any particular meaning. For the use this two vectors is called osculating. of paired data see scatterplots. outliers In statistical data all the values that are Pappus’ theorem Let D be a region of the plane significantly different (too big or too small) than the that lies entirely on one side of a line ` in that plane. most of data are outliers. If D is rotated about `, then the volume of the reoverdamped vibration If a particle oscillates and sulting solid is the product of the area A of D and some additional forces, such as friction, affect its the distance d traveled by the centroid of D. movement, then this kind of vibration is damped vi- parabola One of the three main conic sections. bration and it’s given by the equation Geometrically, a parabola is the location (locus) of all points in a plane that have the same distance from my 00 + cy 0 + ky = 0. a given line (called parametrix) and a point (called focus) not on that line. Equivalently, the parabola In case when the discriminant of the characteristic could be described as the result of cutting a double equation c2 − 4mk < 0, then the vibration is called cone by a plane that is parallel to the generators of overdamped. In this case the characteristic roots are the cone, and not passing through vertex of the cone. negative and the solution has the form Alternative geometric definition could be given with the use of eccentricity. See corresponding definition. −(c/2m)t y=e (c1 cos ωt + c2 sin ωt), Algebraically, the general equation of a parabola is given by the quadratic equation Ax2 + Bxy + Cy 2 + and they are quasiperiodic. Dx + Ey + F = 0, where A, B, C, D, E, F are real overdetermined linear systems A system of constants and A · C = 0. This means that one of the linear equations where there are more equations terms Ax2 or Cy 2 is missing. The case when both than unknown variables. In most cases the overde- A and C are zero is a case of degenerate conic. In termined systems do not have solutions. See also the case when the directrix is parallel to the x-axis underdetermined linear systems. with the equation y = −p and the focus is located overlaping sets Two or more sets that on the y-axis at the point (0, p), the equation of the have common elements. For example, the sets parabola translates into the standard form A = {x| − 5 < x ≤ 2} and B = {x|0 ≤ x ≤ 7} 1 2 x . y= overlap, because the set {x|0 ≤ x ≤ 2} belongs to 4p both of them. The common set is called intersection Similarly, if the directrix is parallel to the y-axes and of two sets and is denoted by A ∩ B. the focus is on the x-axis, the equation will be x = 1/4py 2 . The point where the parabola touches one of the axes is called the vertex. In the more general case when the vertex is shifted, the above equations

106 transform to one of the following: y−k =

1 (x − h)2 , 4p

x−h=

1 (y − k)2 . 4p

if there exists a real scalar c such that u = cv. For parallel vector the angle between them is either 0 or π(180◦ ). parallelogram A quadrilateral where two pairs of opposite sides are parallel. In parallelogram the opposite sides are equal. In the particular case when all four sides have the same size, the parallelogram is called a rhombus.

All of the above cases happen when in the general equation the term Bxy is missing. In the case B 6= 0 the result is still a parabola, which is the result of rotation of one of the previous simpler cases. The parabola could also be given by its polar equation: d d r= or r = , 1 ± cos θ 1 ± sin θ where d > 0.

parallelogram law of vectors.

See addition and subtraction

parameter In mathematics parameters can be describes as something in between constants and variparabolic cylinder The three dimensional surface ables. This means that depending on situation the parameter could either be fixed at some numeric given by (for example) the equation value or start changing its values. In the expression a2 x2 = b2 z. Z ∞ e−λx dx 0

paraboloid

See elliptic paraboloid.

λ is a parameter and x is the variable of integration. For other case where the term parameter is used in slightly different meaning see parametric equations.

parallelepiped A three dimensional solid, similar to rectangular box. Difference is that if for rectangular solid each face is a rectangle, for parallelepiped parametric curve See parametric equation. each face is a parallelogram. parametric equation Many curves that by difparallel lines Two lines on the plane that do not ferent reasons cannot be expressed as a graph of intersect. If two lines are parallel then they have the a function, can be expressed by parametric equasame slope. Example: The lines y = −2x + 1 and tions. Suppose C is a plane curve and the coordiy = −2x − 27 are parallel. nates (x, y) of the curve are given by the functions x = f (t), y = g(t), where f and g are some functions parallel planes Two planes in the space are paralof the variable t on some interval I. The equations for lel if they do not intersect. If a1 x+b1 y +c1 z = d1 and x and y are called parametric equations of the curve a2 x+b2 y +c2 z = d2 are equations of two planes, then C and t plays the role of parameter in this case. they are parallel if and only if a1 /a2 = b1 /b2 = c1 /c2 Examples: (1) The circle given by x2 + y 2 = 1 could but are not equal to d1 /d2 . be written in parametric form as x = cos θ, y = parallel vectors Two vectors u and v are parallel sin θ, 0 ≤ θ ≤ 2π.

107 (2) The parametric equations x = t2 − 1, y = factors. Depending on types of factors we have dift/2, −2 ≤ t ≤ 3 represent a portion of the parabola ferent types of partial fraction decompositions. opening to the right. (1) All the factors are linear and none of them is repeated: (a1 x+b1 )(a2 x+b2 ) · · · (ak x+bk ). In this case parametric surface In three dimensional space, the rational function will have the decomposition the variables (x, y, z) on some surface S sometimes could be written in the form x = f (u, v), y = A2 Ak A1 + + ··· + , R(x) = g(u, v), z = h(u, v), where (u, v) are parameters from a1 x + b1 a2 x + b2 ak x + bk some region G. If this is possible then the surface S is called parametric and the above equations are called where A1 , A2 , · · · , Ak are just constants. (2) All the factors are linear but some of them are parametric equations of the surface. repeated (possibly many times). Suppose the factor parametrization The process of changing regular ax+b is repeated k, i.e., the denominator has the facequation to parametric equation. tor (ax + b)k . Then the decomposition corresponding parentheses The symbols ( ). One of the grouping to this particular factor will have the form symbols along with brackets and braces. Primarily A2 Ak A1 + + ··· + . is used to separate certain numbers and variables to ax + b (ax + b)2 (ax + b)k indicate operations to be done first. partial derivative For functions of two or more The partial fractions corresponding to the nonvariables there is no notion of the derivative but in- repeated linear factors will be as in the first case. stead partial derivatives are considered for each vari- (3) The denominator has non-reducible quadratic fac2 able. Let f (x, y) be a function of two independent tor ax +bx+c. Then the partial fraction corresponding to this particular factor will have the form variables and assume that the limit lim

h→0

Ax + B . ax2 + bx + c

f (x + h, y) − f (x, y) h

exists for any fixed y. Then this limit is called the (4) The denominator has a repeated quadratic facpartial derivative of f with respect to the variable x. tor (ax2 + bx + c)k . Then the corresponding partial (x,y) The notations fx (x, y) or ∂f∂x and some others are fraction will have the form used. The partial derivative fy (x, y) is defined simiA1 x + B1 Ak x + B k larly. The definitions of partial derivatives are similar + ··· + . 2 + bx + c 2 + bx + c)k ax (ax for functions of three or more variables. partial differential equation A differential equation that involves functions of several variables and their partial derivatives. See also heat equation, Laplace’s equation, harmonic functions.

Examples: 1) To expand the rational function

partial fraction expansion Any rational function of the form R(x) = P (x)/Q(x), where P and Q are polynomials, could be written in the form of the sum of simpler rational functions, called partial fractions. In order to find this decomposition, as a first step we need to assure that the fraction is proper and if it is not, perform the division and deal with the remainder only, which is necessarily proper. On the next step we factor the denominator into linear and quadratic

we write

f (x) =

x2

x+7 x+7 = −x−6 (x − 3)(x + 2)

x+7 A B = + , (x − 3)(x + 2) x−3 x+2 where A, B are yet to be determined coefficients (this is called the method of undetermined coefficients). Multiplying both sides by the common denominator we get x + 7 = A(x + 2) + B(x − 3) and expanding and equating coefficients of similar powers we get a

108 system of two linear equations with unknowns A and partial sum of a series For an infinite series P∞ a the sum of the first n elements Sn = B: k Pk=1 n a . Here the terms ak may be constants or A+B =1 k k=1 variable terms such as powers of the variable x. 2A − 3B = 7 particular solution For non-homogeneous differThis system has solution A = 2, B = −1 and we ential equations of the type have −1 2 + f (x) = y 00 + p(x)y 0 + q(x)y = g(x) x−3 x+2 which is the partial fraction decomposition for the is any solution of this equation. The importance of function f (x). particular solution is that the general solution of this 2) The function linear equation is the sum of that particular solution and the general solution of the corresponding homo5x2 + 20x + 6 geneous equation f (x) = x(x + 1)2 y 00 + p(x)y 0 + q(x)y = 0. should have the expansion partitioned matrix A matrix that is divided into B C A smaller parts, usually called blocks. A matrix could + f (x) = + . x x + 1 (x + 1)2 be partitioned in many different ways depending on Again, multiplying both sides by the common denom- the need. inator and equating the coefficients we will get a system of three linear equations with unknowns A, B, C. Solving the system we will have A = 6, B = −1, C = 9 and the decomposition f (x) =

6 −1 9 + + . x x + 1 (x + 1)2

Pascal’s triangle A triangular table of binomial coefficients for easy calculation of binomial expansion. Each entry at the ends of this table is 1, and any other entry is the sum of two entries right above itself. 1 1

3) For the function

1

8x2 + 13x f (x) = (x2 + 2)2 we have the expansion f (x) =

Cx + D Ax + B + 2 x2 + 2 (x + 2)2

1

1

1 2

3

1

4

5

10

1 3

6

1 4

1

10

5

1

························

Example: If we want to expand (x + y)4 , then we use and the same method as before gives A = 8, B = the numbers on the 5th row and get 0, C = −3, D = 0. Now, the expansion is (x + y)4 = x4 + 4x3 y + 6x2 y 2 + 4xy 3 + y 4 . 8x2 + 13x 8x −3x = 2 + 2 . path Any curve could be called a path. Most com2 2 (x + 2) x + 2 (x + 2)2 monly the term is used to indicate a piecewise-smooth See also integration by partial fractions. (differentiable) curve. partial integration See integration by parts.

pendulum equation

An equation that describes

109 the movement of a pendulum. Depending on assumptions, the form of the equation may vary. If a pendulum consists of a mass M that is hanging on one end of a weightless rod of the length L and the angle of displacement from the vertical position is denoted by θ, then the movement of the pendulum is described by the equation dθ dθ +γ + ω 2 sin θ = 0. (1) dt2 dt Here ω 2 = g/L, where g is the gravitational constant and the constant γ depends on the damping force acting on the mass m. This equation is called nonlinear damped pendulum equation. For small θ this equation could be linearized by substituting θ instead of sin θ and the corresponding equation

100 values and even for large sets it is impossible to divide the set into 100 exactly equal parts. See also quartiles. perfect numbers A number is called perfect if it is the sum of all of its positive divisors except the number itself. 6 is perfect because it has divisors 1, 2, 3 and 1+2+3=6. Also 28, 496 and 8128 are perfect numbers. There are only a few perfect numbers known and it is not known if there are finite or infinite perfect numbers.

perimeter Usually, the length of a closed curve. In some cases just means the curve itself. To find the perimeter of some geometric figure means to find the length of the curve that bounds that geometric figure. The perimeter of any polygon is calculated by simply adding the lengths of segments that form the polygon. For example, if the polygon is a quadrilatdθ dθ 2 +γ +ω θ =0 (2) eral, then the perimeter is the sum of lengths of all dt2 dt four sides. In cases more complicated than a polyis called linear damped equation. If the damping gon, usually only the use of calculus allows to find coefficient γ (and the corresponding term in (1) or the perimeter. See arc length for details. (2)) is removed, then equation (1) becomes nonlinear undamped and (2) becomes linear undamped. In period Let the function f has the property that this last case the solution of the resulting equation there exists positive real number m such that θ00 + ω 2 θ = 0 is the function f (x + m) = f (x) θ(t) = A cos ωt + B sin ωt for all x in the domain of f . The smallest such and is called simple harmonic motion. Its period is number m is called the period of the function f and the number T = 2π/ω. the function itself is called periodic with period m. The functions sin x, cos x both have period 2π and percent One hundredth part of a number. The nothe functions tan x, cot x have period π. tation n% means n/100. Examples: 27% = 27/100 = 0.27, 436.7% = 436.7/100 = 4367/1000 = 4.367. period of simple harmonic motion See pendulum equation. percentile Suppose we have a large set of numeric values organized in increasing order. We can divide periodic function See period. this set into hundred parts by putting ”almost equal permutation For a given set of integers number” of values in each part. If we denote the {1, 2, · · · , n} any rearrangement of the numbers first point of division by P1 , second point of division without repeating or omitting any of them, is called by P2 and so on, then we will get 99 points of divia permutation of that set. Any permutation is sion P1 , P2 , · · · , P99 . The points between the smallest a combination of finite number of transpositions. value and P1 belong to the first percentile, the points transposition is a simplest permutation when the between P1 and P2 – to the second percentile, and so places of only two numbers are interchanged. See on, until we get to the 100th percentile which consists also odd and even permutations. of values between P99 and the largest value in the set. Obviously, this is possible only if the set has at least perpendicular lines Two lines in the plane are

110 perpendicular if they intersect and form a right defined differently on different parts of its domain. (90◦ ) angle. The slopes of perpendicular lines are The function opposite reciprocals: If m1 and m2 are the slopes of 2 these lines then m1 · m2 = −1. Example: The lines f (x) = x − 1 if x ≥ 0 2x + 5 if x < 0 y = 2x − 1 and y = − 12 x + 4 are perpendicular. is an example of piecewise defined function. perpendicular vectors See orthogonal vectors. plane One of the basic objects of Euclidean geometry along with points and lines. In Cartesian coordinate system planes could be given by equations. The most general equation of the plane is written in the form ax + by + cz = d where a, b, c d are any real numbers. Any plane could be uniquely determined by three points (according to one of the Euclidean postulates). The three coordinate planes (xy, xz, yz) are given by the equations z = 0, y = 0 pie chart One of the ways of visualizing data, and x = 0 respectively. Any plane could be deterusually qualitative data along with bar graphs. The mined also by any point on that plane (P0 (x0 , y0 , x0 , picture below shows a data that could be put in for example) and the normal vector to that plane. five categories and the size of the corresponding Suppose P (x, y, z) is an arbitrary point on the plane shaded area shows the proportion of the values in and r and r0 are the vectors corresponding to the that category. points P and P0 . Then, if n is the normal vector to the plane at the point P0 , it is perpendicular to the vector r−r0 and, as a result, their dot product is zero. That relationship is the determining equation of the plane and is given by the equation n · (r − r0 ) = 0. This is called the vector form of the equation of the plane. On the other hand, if the equation of the plane is ax + by + cz = d, phase shift For trigonometric functions. If the argument of the function y = sin t is shifted by adding or subtracting a constant, such as in y = sin(t − a), then this number a is the phase shift of the function. Geometrically this means (for a > 0) that the graph of the function is shifted to the right (”delayed”) by a units. The same definition is true for all other trigonometric functions.

then the normal vector n has the components (a, b, c). The tangent plane to a surface at some point P is a plane that passes through that point and is perpendicular (orthogonal) to the normal vector to the surface at that point. If the equation of the surface is given by the function z = f (x, y), then the equapiecewise continuous function A function de- tion of the tangent plane to that surface at the point fined on some interval I (finite or infinite) that is P0 (x0 , y0 , z0 ) is given by the equation continuous everywhere on the interval except finite ∂f ∂f (x − x0 ) + (y − y0 ). z − z0 = number of points (in case of bounded interval) or in∂x ∂y finitely many points with no limit points inside the interval (in case of unbounded interval). The saw- Similar equation is valid also for surfaces given by the tooth function, and the Heaviside function are exam- equations F (x, y, z) = k. ples of piecewise continuous functions. point The most basic geometric object defined to piecewise defined function A function that is have no measurements. Points on the line, plane, or

111 space could be given by their coordinates. Depending on situation we have one, two, or three coordinates respectively. For example, the point with the coordinates (1, 2, −3) is point in three-dimensional space and the point (−2, 5) is on the xy-plane. point of inflection See inflection point. point-slope equation of the line If the slope m and a point (x1 , y1 ) through which a line passes are known, then the equation of that line is given by the formula y − y1 = m(x − x1 ).

any integer n because geometrically they represent the same point. Additionally, the same point could be written as (−2, 5π/4) because this means that we move the angle 5π/4 radians (and this is opposite to the angle π/4) and then move 2 units backwards along the side of this angle. Additionally, the same point now could be written as (−2, 5π/4 + 2πn). The picture below shows the presentation of two different points with angle measured in degrees.

See also line. Poisson distribution A probability distribution describing certain real life situations. If the distribution has the mean λ and x denotes the number of ”successes”, then the probability P (X = x) =

e−λ λx . x!

The √ standard deviation of the Poisson distribution is λ. polar coordinates Along with Cartesian coordinate system, the most important method of representing points on the plane. The system consists of a point (the analog of the origin) called the pole and a ray coming out from the pole, called polar axis. By convention polar axis is drown horizontally going to the right. To represent a point on the plane in polar coordinates we measure the distance from the point to the pole (usually denoted by r) and measure the angle formed by the polar axis and the line segment connecting the point and the pole (usually denoted by θ). The angle could be measured by either the radian measure or degree measure. Also the ”distance” is allowed to be both positive or negative (see explanations below). This way we put each point into correspondence with a pair of real numbers as in the case of Cartesian system. The major difference is that while to each pair of real number (r, θ) there is exactly one corresponding point, the points themselves could be represented in infinitely many ways. For example, the point (2, π/4) could be also written as (2, π/4 + 2πn) for

There are simple relations between rectangular and polar coordinates of a point. If the polar coordinates are known, then rectangular coordinates are found by the formulas x = r cos θ,

y = r sin θ.

Conversely, if the rectangular coordinates are known, then y r2 = x2 + y 2 , tan θ = . x This second pair of equations show why the polar representations are not unique. The ”radius” r has p two possible values r = ± x2 + y 2 and the angle θ has infinitely many possible values (differing by an integer multiple of π). polar equations Equations with respect of the polar coordinates, most generally written in the form F (r, θ) = 0. In practice, however, the equations are usually written in a simpler form r = f (θ). Many plane curves that are difficult (and sometimes impossible) to write in rectangular system have fairly simple representations in polar coordinates. Additionally, many equations that are not functions in rectangular system turn out to be simple functions in polar coordinates. For example, the equation of

112 the circle centered at the origin with the radius a x2 +y 2 = a2 which is not a function, could be written as a simple function r = a in polar coordinates. Here are other examples of curves in rectangular and polar coordinates: Line y = ax, θ = a, circle with center (1, 0) and radius 1: (x − 1)2 + y 2 = 1, r = 2 cos θ. Most of the polar curves are usually difficult to represent and especially graph in rectangular coordinates. For examples of more polar curves and their graphs see the entries cardioid, lima¸con, four leaf rose and others. polar form of a complex number See trigonometric form of a complex number. polygon A geometric figure made up of three or more segments of a straight line, connected at the endpoints. Polygon with three sides is called a triangle, with four sides - a quadrilateral, with five sides a pentagon, and so on. See also regular polygon. polynomial The sum of finite number of combinations of power functions. Polynomials, as a rule, are written in the decreasing order of powers, but in some cases it is convenient to write them in increasing order. The general form of the polynomial of degree n is

considered positive if the terminal side is achieved by moving counterclockwise starting from the initial side. See more details in the article angle. positive definite form A quadratic form is positive definite if it is positive for all values of variables xj except when all of them are zero: n X

aij xi xj > 0

i,j=1

unless x1 = x2 = · · · = xn = 0. If the quadratic form is represented by a matrix A then the corresponding matrix is called positive definite. See also negative definite matrix. positive definite function A function F (x, y) defined on some domain D containing the origin is positive definite, if F (0, 0) = 0 and F (x, y) > 0 everywhere else in D. If F (x, y) ≥ 0, then the function is positive semidefinite. See also negative definite function.

p(x) = an xn + an−1 xn−1 + · · · + a1 x + a0 .

positive definite matrix A symmetric matrix A is negative definite, if for any vector x 6= 0, xT Ax > 0. If the inequality is substituted by ≥, then the matrix is positive semidefinite. Here xT indicates the transpose of the column vector x. See also negative definite matrix.

The polynomial p(x) = 5x4 − 3x2 + 2x + 4 is a polynomial of degree 4.

positive function A function that takes only positive values.

pooled estimator In statistics, when comparing two proportions, means, variances, etc., often the values from two populations are ”pooled” together and the resulting numeric values are called pooled estimators. Example: Suppose we need to compare proportions of two populations with the following information. Sample sizes are are n1 and n2 respectively and the numbers of ”successes” are m1 and m2 . Accordingly, the proportions would be p1 = m1 /n1 and p2 = m2 /n2 . In this notations, the pooled success proportion would be

positive number zero.

ppooled =

m1 + m2 . n1 + n2

positive angle An angle in standard position is

A number that is greater than

positive orientation of a simple closed curve is agreed to be the counterclockwise direction as we move along the curve. positive orientation of a closed surface is agreed to be the one for which the normal vector points outward from the region enclosed in surface. power function The function f (x) = xn , where n is a whole number, n = 0, 1, 2, · · ·. power law of limits positive integer n,

For any function f (x) and

h in lim [f (x)]n = lim f (x) .

x→a

x→a

113 In the particular case when f (x) = x we have are solutions of that equation then so is their sum limx→a xn = an . function y1 + y2 . power series A series formed by the combinations of power functions:

probability (1) One of the main branches of mathematics dealing with events which outcome is governed by chance. c0 + c1 x + c2 x2 + c3 x3 + · · · + cn xn + · · · (2) Probability of an event is the ”likelihood” that that event will happen. It is measured by a numeric The numbers c0 , c1 , c2 , · · · are called the coefficients value that varies between 0 and 1. The event that of the power series. Each power series converges on cannot happen (an impossible event) has probability some interval of the form −R < x < R, where R is zero and an event that is certain to happen has probcalled the radius of convergence. Depending on coef- ability one. For discrete variables the probability of ficients, the radius R may be zero (series converges some event A can be calculated by the formula only at x = 0), infinite (series converges for any real number of times A happened x), or finite. In this last case the series may or may , P (A) = number of total experiments not converge at the endpoints x = −R or x = R (see also radius of convergence). The power series could where by ”experiment” we understand observations be differentiated or integrated term-by-term just as we produced to see if A happens. The simplest exany polynomial. In the more general case, also power ample is tossing a coin multiple times and observing series centered at and arbitrary point x = a are conhow many times we get the ”tail” (the event A). sidered: If the number of outcomes is some fixed number, such as in the case of tossing a coin (exactly two possible c0 + c1 (x − a) + c2 (x − a)2 + c3 (x − a)3 + · · · outcomes: ”head” or ”tail”), or rolling a dye (six This series has the exact same properties as the series possible outcomes: the values from 1 to 6), then the probability of an event could be calculated by the centered at zero. formula prime number A natural number that cannot be number of ways A happens divided evenly by any other natural numbers except P (A) = . number of total outcomes 1 and that number itself. All the other natural numbers are called composite. The number 1 is considered For operations with probability values see multiplineither prime nor composite. First few prime numcation rule for probabilities, addition rule for probbers are: 2,3,5,7,11,13,17,19,23,29,31... By the Funabilities. For probability distributions see the entry damental theorem of arithmetic any natural number distribution and the related entries mentioned in that is the product of prime numbers. article. See also conditional probability. principal nth root Let a be a real number that probability model A mathematical model where has at least one nth root. The principal nth root of the numeric values used to construct that model are a is √ the one that has the same sign as a and denoted √ probabilities of some events. For examples see binoby n a. For a = 4 the number 4 = 2 is the princimial distribution, Poisson distribution, normal dispal square root because it is positive. For a = −27 √ tribution. 3 the number −27 = −3 is the principal cubic root probability density function A function that is because it is negative. determined by the probabilities of some event(s). The principle of mathematical induction See mathdefining properties of these functions are: ematical induction. (1) The function is always non-negative; principle of superposition for linear homoge- (2) The area under that function is exactly one. neous differential equations states, that if y1 and y2 See also entries related to specific probability models.

114 product The result of multiplication of two (or more) objects. The objects may be real or complex numbers, functions, matrices, vectors, etc. For specific definitions in all of these cases see the entries multiplication of complex numbers, of fractions, of functions, of matrices and also cross product, dot product, scalar product. product formulas Also called product-to-sum formulas. In trigonometry, the formulas sin x sin y =

1 [cos(x − y) − cos(x + y)] 2

7 12

we cross-multiply and get the equation 12x = 7 · 3 and x = 7/4. P-value Also sometimes called probability value. One of the methods of hypothesis testing. The Pvalue is the probability of observing a value like the given value (or even less likely) assuming that the null hypothesis is true. The smaller the P-value the more evidence is there to reject the null hypothesis. Big Pvalues indicate that there is not enough evidence to reject the null hypothesis (so we have to accept it). Visually, the P-value is the area under the normal probability curve from the observed vaue to infinity (in the case of right-tailed test), the area from the observed value to minus infinity (left-tailed test), or the sum of areas in both directions (two-tailed test).

1 [cos(x + y) + cos(x − y)] 2 1 sin x cos y = [sin(x + y) + sin(x − y)] Pythagorean identities The trigonometric iden2 tities 1 sin2 θ + cos2 θ = 1 cos x sin y = [sin(x + y) − sin(x − y)]. 2 tan2 θ + 1 = sec2 θ Similar formulas are also possible for other functions but hardly ever used. See also trigonometric identi1 + cot2 θ = csc2 θ. ties. See also trigonometric identities. product rule For differentiation. If the functions Pythagorean theorem One of the most famous f and g are differentiable, then theorems of geometry that states that the square of the length of the hypothenuse of any right triangle is [f (x)g(x)]0 = f (x)g 0 (x) + g(x)f 0 (x). equal to the sum of the squares of the lengths of the two legs. Symbolically, if a, b represent the lengths of Examples: (1) (x2 sin x)0 = 2x sin x + x2 cos x, legs and c is the length of the hypothenuse, then (2) (tan x ln x)0 = sec2 x ln x + 1/x tan x. cos x cos y =

projection The mapping of a point onto some line or plane. The most common projections are the orthogonal projections. See the corresponding entry for the details and examples. proper value

See eigenvalue.

proportion A statement that two or more ratios are equal. Ratios may contain just numbers or variables or even functions. Examples of proprtions are 3 6 x 7 5 = 10 , 3 = 12 or the Law of Sines: sin α sin β sinγ = = . a b c To solve a proportion means to find the unknown quantity. For example, to solve the proportion x3 =

a2 + b2 = c2 .

115

Q

x1 , x2 , · · · , xn is the expression of the form n X

quadrants The x−axis divides the Cartesian plane into upper and lower half-planes. The addition of the y−axis divides the plane again and as a result four quadrants are formed. The quadrants are counted starting with the one corresponding to inequalities x > 0 and y > 0 and moving against the clock.

aij xi xj .

i,j=1

For the case of two variables x and y the quadratic form is ax2 + by 2 + cxy. quadratic formula See quadratic equation.

quadratic function Also called quadratic polyno2 quadrantal angle Any angle in standard position mial. The function of the form f (x) = ax + bx + c where the terminal side coincides with any of the co- where a, b, c are numeric coefficients (real or comordinate axes. Hence, quadrantal angles can measure plex) and x is the variable. The graph of the 0◦ , 90◦ , 180◦ , 270◦ or the same sizes plus any mul- quadratic function is a parabola. Every quadratic tiple of 360◦ in degree measure. In radian measure function could be written in the standard form f (x) = a(x − h)2 + k, where the point (h, k) is the quadrantal angles measure 0, π/2, π, 3π/2. coordinate location of the vertex of the parabola. quadratic approximation Approximation of a given function by a quadratic function (polynomial). quadratic inequality An inequality that involves Taylor and Maclaurin polynomials of second degree a quadratic function. To solve quadratic inequality represent examples of quadratic approximation. For means to find all the values of the variable x that example, for the exponential function f (x) = ex substituted into the inequality result in a true state2 the Maclaurin polynomial T2 (x) = 1 + x + x2 is its ment. Solutions of quadratic inequalities are given either in the form of a finite interval (open or closed) quadratic approximation near the origin. or in the form of the union of two semi-infinite inquadratic equation The equation tervals. Examples: The inequality x2 − x − 6 > 0 has the solution set (−∞, −2) ∪ (3, ∞). The inequality 2x2 − 2x − 4 ≤ 0 has the solution set [−1, 2]. ax2 + bx + c = 0. Some inequalities have no solution (have empty solu2 This equation always has solution(s) given by the tion set), such as the inequality x + 1 < 0, because 2 the quadratic expression x + 1 is always positive. quadratic formula: x=

−b ±

√

b2 − 4ac . 2a

The expression under the square root sign is called the discriminant and it determines what kind of zeros (solutions, roots) this equation has. (1) The equation 2x2 − 3x − 4 = 0 has, by the quadratic formula, two √ distinct real solutions x = 34 ± 441 . (2) The equation x2 − √ 2x + 3 = 0 has two distinct complex roots x = 1 ± i 2 and (3) the equation x2 − 2x + 1 = 0 has two repeated roots x = 1. For another method of solutions of quadratic equations see also factoring quadratic forms

Quadratic form of n variables

quadratic surface A three dimensional surface given by a quadratic polynomial which consists of a quadratic form plus some linear and constant terms such as x2 − 2y 2 + z 2 + xy − xz + 3yz − x − 5y + 2z = 8. See also ellipsoid, hyperboloid, paraboloid. quadrilateral Also called quadrangle. A geometric figure that consists of four connected interval of a straight line. Among the most common quadrilaterals are the square, rectangle, parallelogram, rhombus, trapezoid.

116 quartic equation Algebraic equation of the fourth degree that could be written in the form αy 4 + βy 3 + γy 2 + δy + = 0,

α 6= 0.

There is a formula for solving these types of equations which is extremely complicated and difficult to use in practice unlike the quadratic formula (see also cubic equation). Here is a quick outline of the solution process. First, we divide the equation by α 6= 0 and get a monic equation qualitative data Also called categorical data. Data that is organized by its qualitative (as opposed to quantitative) properties. This kind of data may give information in the non-numeric form, such as eye color, agree-disagree opinions, month of the year, etc. It also may contain numeric information put into categories, such as year of birth, ZIP codes and others, that have no particular significance of order. qualitative variable A variable that takes qualitative values only. quantitative data Numeric data with added condition that the order of numbers has significant importance. The numeric data of heights is quantitative because people of different heights can be put in specific order (by increasing or decreasing heights). The numeric data of birthdates is not necessarily quantitative because we cannot organize people in order of birthdates (the year of birth is also necessary to do so). quantitative variable A variable that takes quantitative (numeric) values only. quartiles Suppose we have a set of numeric values organized in increasing order. We first find the median of this set (see corresponding definition for the procedure). On the second step we additionally find the medians of the two halves. This way the set of values is divided into four equal (or almost equal) parts. Each of them is called a quartile. The notations for the three points dividing the set are Q1 , Q2 , Q3 . Note also that second point Q2 coincides with the median and usually is denoted by M . Finding the quartiles is important for constructing boxplots.

y 4 + ay 3 + by 2 + cy + d = 0, then by the substitution y = x − a/4 bring it to a simpler form x4 + px2 + qx + r = 0,

(1)

where the new coefficients p, q, r are found from the old coefficients a, b, c, d. Now we add the expression 2zx2 + z 2 to both sides of this equation and bringing some terms to the right side, get x4 + 2zx2 + z 2 = (2z − p)x2 − qx + (z 2 − r), where z is yet to be determined. The left side is the perfect square (x2 + z)2 and the right side could be made perfect square if we chose z to satisfy the condi√ √ tion 2 2z − p z 2 − r = −q. Squaring this relation gives (2z −p)(z 2 −r) = q 2 /4 which is a cubic equation with respect to the unknown z: p pr q 2 z 3 − z 2 − rz + − = 0. 2 2 8 This equation could be solved by the method described in the article cubic equations and the four solutions of the equation (1) are given by the formulas r p 1 1p 1 2z − p ± − z − p + z 2 − r, x1,2 = 2 2 4 r p 1 1 1p 2z − p ± − z − p − z 2 − r. x3,4 = 2 2 4 For equations where some more terms are missing the solution formula becomes slightly easier. In practice,

117 approximate methods are more appropriate and easier to use than the formulas above.

R

quasi frequency Some functions are not periodic but have certain repeating pattern. Example: The function f (x) = e−x sin 2x is not periodic but it oscillates with specific frequency. In this case it is two. The number 2 is the quasi frequency of the function f . The number 2π/2 = π in this case will be the quasi period of the same function.

radian measure One of the two main units used to measure angles, especially useful for applications in calculus. One radian, by definition, is the central angle that is formed when a segment equal to the radius of the circle is placed on the circumference of the circle. If we place a segment of the length s on quotient of numbers The same as the ratio of two the circumference, the radian measure of the correnumbers. The result of the division of two numbers. sponding central angle will be equal θ = s/r, where 5 r is the length of the radius. Since the length of the 7 is the quotient of numbers 5 and 7. quotient rule For differentiation of the quotient circle is 2πr, the radian measure of a whole circle will of two functions. If f and g are two differentiable be 2π. See also degree measure. functions on some interval I, then radicals Also called roots. The second meaning of √ n a indicating the n-th root the term is just the sign 0 f 0 (x)g(x) − g 0 (x)f (x) f (x) of the number a. See the entry roots for all the de= . g(x) [g(x)]2 tails. See also differentiation rules.

radical equation Equation that contains the unknown variable under radical of any √ √ order. Examples could be 2x + 7 − x = 2 or 3 3x − 1 = 2. The standard method of solving most of the radical equations is to isolate the radical expression and then raise both sides of the equation to the power of the radical. When solving radical equations containing even order radicals (such as square roots or forth degree roots) this method may produce extraneous roots. To avoid getting incorrect roots all the solutions should be checked back into the original equation to assure the validity of that solutions. For example, to solve the equation in our first example, we move the variable x to the right side to isolate the radical first: √ 2x + 7 = x + 2 and then squaring both sides get the algebraic equation 2x + 7 = x2 + 4x + 4. This quadratic equation has two zeros x = 1, x = −3. After checking in the original equation both of these solutions we see that x = −3 is an extraneous solution and the only valid solution of the equation is x = 1. To solve the second example we just raise both sides to the third power (cube both sides) and get a simple linear equation 3x − 1 = 8 with a single solution

118 x = 3. Extraneous solutions do not arise in this case radius of convergence For a given power series or in any case where only odd order radicals are in∞ X volved, so x = 3 is a valid solution. cn (x − a)n Solutions of some radical equations require repeated n=0 use of the method described above. For example, to solve the equation there are three possibilities for convergence: √ √ (1) The series converges only for x = a; 2x − 5 − x − 3 = 1 (2) The series converges for all real x; we first move the second radical to the right and (3) There is finite number R > 0 such that the series converges for all |x − a| < R and diverges for all square both sides to get |x − a| > R. √ In the third case we say that the series has finite ra2x − 5 = x − 3 + 2 x − 3 + 1 dius of convergence equal to R. The interval −R+a < which still√contains a radical. Now we can isolate it, x < R + a is called the interval of convergence. At x − 3 = 2 x − 3 and squaring the second time get the endpoints x − a = ±R of the interval of converthe quadratic equation x2 − 10x + 21 = 0 with two gence the series may or may not converge. In the solutions x = 3, x = 7. This time both solutions first and second cases we say that the radius of conturn to be valid. vergence is zero or P is infinite respectively. Examples: ∞ n (1) For the series radicand A number or an expression appearing n=0 n!x , R = 0; (2) For the P n √ ∞ x under the radical. In the expressions 5 7 and series n=0 n! , R = ∞; (3) For the series √ 5 3 2 4x − 3x + x − 1, 7 and 4x5 − 3x3 + x2 − 1 are ∞ X (−1)n xn the radicands respectively. n+1 n=0 radiocarbon dating Or just carbon dating. A method of determining the age of old objects using the properties of radioactive carbon C 14 and the R = 1. This series converges at the right endpoint of the interval x = 1 and diverges at the left endpoint knowledge of its half-life period. Suppose we need to determined the age of some an- of the interval x = −1. cient artifact that has remaining amount of carbon-14 random digit A number (usually whole number at the amount A. The original amount of the carbon between 0 and 9) that is chosen as a result of the is always the same (in percentages) and is denoted by chance. Tables of random digits are sometimes used A0 . The age of the artifact can now be determined in the process of randomization of statistical experiby the formula ments. T = −8267 ln(A/A0 ).

radius of a circle By definition, a circle is the geometric place (the locus) of all points that have equal distance from a given fixed point. This distance is the radius of the circle. In the general equation of the circle (x − a)2 + (y − b)2 = r2 the points (x, y) represent an arbitrary point of the circle, the point (a, b) is the center, and r is the radius.

random number The same as random digit with the difference that chosen numbers are not necessarily digits. random variable A variable that assumes its values randomly (by chance). One of the main objects of Statistics. range of a function The set of all values that the given function y = f (x) can take. Examples: For the function f (x) = 2x5 − 3x2 + 5x − 1 the range is the set of all real numbers; for the function f (x) = 2x the range is the set of all positive numbers.

119 range of a transformation For a linear transformation T : V → W the set of all vectors w ∈ W such that there is at least one vector v ∈ V that T v = w. These values in the range are also called images of the transformation T .

rational equation An algebraic equation involving just rational functions. The general form of these type of equations is P (x)/Q(x) = 0 where both P and Q are polynomials. The standard method of solving these kind of equations is to multiply both sides of the equation by the denominator function Q(x) and rank (1) For the linear transformation T : V → W get a polynomial equation P (x) = 0. The only differthe rank is the dimension of the range of that transence compared to polynomial equations is that after formation. finding the solutions of the equation P (x) = 0 we (2) For a matrix A the row space and column space need to make sure that there are no extraneous zeros have the same dimension. This dimension is called (roots). To do that we need to plug in into the origthe rank of the matrix. inal equation all the solutions and see if all of them rate of change The measure of change of some really satisfy that equation. Examples: variable quantity in a unit amount of time. Depend- 1) Solve the equation ing of the nature of change the rate might be a con3 2 stant or a variable amount itself. For objects moving = − 1. x x − 2 with constant speed the rate of change is just their velocity. For objects that move at a variable speed Multiplying by the common denominator x(x − 2) we the average rate of change in a given time interval is get the quadratic equation 2(x − 2) = 3x − x(x − 2) defined to be the difference of their positions divided or, after simplification, x2 − 3x − 4 = 0 which has by the time elapsed: vav = (s2 − s1 )/(t2 − t1 ). For two solutions x = 4, x = −1. Substituting into the these type of moving objects the instantaneous rate original equation we see that both are solutions of the of change is the limit of the above expression as the equation. time interval approaches zero: 2) When we solve the equation s2 − s2 . v = lim 5 x+1 t2 →t1 t2 − t1 2+ = x−4 x−4 This last limit is just the derivative of the position function s = s(t). by the exact same method, we get a linear equation rate of growth (decline) In differential equation with the solution x = 4. Now, if we put this solution dy/dt = ry the constant r is sometimes called rate of back into the original equation then it will become growth or decline depending on its sign (positive or undefined because both denominators become zero. As a result we conclude that the equation has no sonegative). lution. ratio The result of the division of two numbers or expressions. Ratio of two numbers is more general rational function A function that could be reprethat a rational number, because we can divide any sented as a ratio of two polynomials. The domain of two numbers, not just integers. Hence, 3/4 is the a rational function is the set of all real numbers where the denominator does not equal zero. The functions ratio of the numbers 3 and 4 and also is a rational √ √ number, but 3/4 is the ratio of the numbers 3 2x3 − 3x2 + 1 x−1 , and 4 which is not rational (it is irrational ). The 2 x −4 x2 + 1 expression 2x2 − 3x + 1 are rational. The first one has the domain {x|x 6= x3 − 4 ±2} and for the second one the domain is the set of is the ratio of two algebraic expressions (polynomi- all real numbers, because denominator x2 + 1 is never als). equal to zero.

120 rational number A real number that could be Examples: (1) In the expression represented as a ratio of two integers: r = n/m, 2 where m 6= 0. Rational numbers can be repre√ 5x sented also as decimals. The decimal representation of rational numbers is either a terminating decimal we can multiply both numerator and denominator by or non-terminating, repeating decimal. Examples: √ 5x and get (assuming x ≥ 0) 137 2 5 , −218 , −2.759, 17.247247247.... √ √ 2 2 5x 2 5x rational root test A method of finding roots (ze√ =√ = 5x ros, solutions) of a polynomial equation of order 3 or 5x 25x2 higher. This method does not give a guarantee that these roots will be found, just allows to find them in which has no radical in denominator. (2) s cases the equation has rational roots. p r 3 2 3x 12xyz 2 3 4yz · 3x 3 = = . 2y 2 z 8y 3 z 3 2yz Let f (x) = an xn + an−1 xn−1 + · · · + a2 x2 + a1 x + a0 be a polynomial of degree n ≥ 1 with rational co- (3) In situations when the denominator contains the efficients.Then all the rational roots of the equation sum or difference of two square roots of the form √a+ f (x) = 0 (if they exist) have the form p/q, where p √b or √a − √b, the standard method of rationalizing is a factor of the constant term a0 and q is a factor is to multiply both numerator and denominator by of the leading term an . the conjugate of the denominator: Examples: For the equation 2x3 +3x2 −8x+3 = 0 the possible rational roots are ±1, ±3, ± 12 , ± 23 . Checking these values we see that x = 1, 1/2, −3 are the roots of the equation. For the equation x4 −4 = 0 the rational root test shows that the only possible rational roots are ±1, ±2, ±4. The check of all these values shows that none of them are roots of the equation. Instead, the simple factoring x4 − 4 = (x2 + 2)(x2 −√ 2) = 0 shows that the equation has four roots x = ± 2 and √ x = ±i 2. First two are irrational and the second two are complex. The equation has no rational roots.

√ √ 1 7+ 5 √ = √ √ √ √ √ 7− 5 ( 7 − 5)( 7 + 5) √ √ √ √ 7+ 5 7+ 5 = = . 7−5 2 ray Take a line and consider only the part that starts at some (arbitrary) point and goes in one direction. The result is a ray.

real axis The same as x-axis. The term is mainly used when considering representation of complex ratio test A method of determining the absolute numbers as points on the plane. In that situation convergence or divergence of a (numeric or func- the plane will be called complex plane and the y-axis tional) series. will be called imaginary axis. (a) If limn→∞ aan+1 = a < 1, then the series n real line The same as real axis or x-axis. The only P∞ a is absolutely n n=1 difference is that we speak of real line mainly in cases convergent. when we want to represent real numbers as opposed (b) If limn→∞ aan+1 = a > 1, then the series is di n to representing complex numbers or ordered pairs. vergent. (c) In the case a = 1 the test cannot give a definite Each real number corresponds to some point on the real line and any point on the real line represents answer. some real number. rationalizing denominators In some algebraic expressions the denominator contains radical expres- real number The set of all real numbers is the sions. The process of getting rid of that irrationality union of all rational and irrational numbers. In turn, these subsets have their own subsets. The subsets of is called rationalizing the denominator.

121 rational numbers are prime, natural, whole, integer numbers and the subsets of irrational numbers are algebraic and trascedental numbers. Each real number could be represented as a point on the real line and each point on the real line represents a real number. These last statements say, in particular, that inside the real numbers there are no ”gaps”. real part of a complex number In the complex number z = x + iy the number x, which is real. Example: In the complex number −2 + 3i the real part is −2. The number 3 here will be the imaginary part of that complex number.

any term of the sequence (starting from a certain point) it is necessary to know the previous term or terms. Examples: The sequence {an } given by the formula a0 = 2, an = 2 · an−1 for n ≥ 1 requires the knowledge of only the previous term. In the Fibbonaci sequence a1 = a2 = 1, an = an−1 + an−2 for n ≥ 3 we need to know the two previous terms to determine the next one.

reduced row-echelon form Similar to rowechelon form with additional fourth condition: (4) Any column that contains a leading 1 has zeros everywhere else. The matrices rearrangement of series Any change in order of 1 0 0 1 0 0 −2 terms of a series is called rearrangement. Generally 0 1 0 0 1 0 3 . speaking, for infinite series, rearrangement changes 0 0 1 0 0 1 2 the sum of the series. The following are two important statements about rearrangements of the numeric P∞ are in reduced row-echelon form. series n=0 an . (1) If the series is absolutely convergent, then any re- reduction formula Any formula that allows to rearrangement of the series leaves the sum unchanged. duce an expression to a simpler form. Many trigono(2) If the series is conditionally convergent and L is metric formulas could be considered reduction formuany real number, then it is possible to rearrange the las (double angle, half angle, etc). Also, many integral series in such a way that its sum becomes equal to L. formulas have the name reduction formula. Example of an integral reduction formula is: reciprocal function Most commonly used to inZ Z dicate the function f (x) = 1/x. (ln x)n dx = x(ln x)n − n (ln x)n−1 dx. reciprocal number Or reciprocal of a number. For a given number a, a 6= 0 the reciprocal is the number that is multiplied by a gives 1. Hence the reduction of order A method of finding solutions reciprocal of a will be a1 . Examples: The reciprocal of differential equations (with constant or variable coof −2 is −1/2 and the reciprocal of 3/7 is 7/3. efficients) when one of the solutions of that equation reciprocal trigonometric identities The identi- is known. The method allows to use that solution and reduce the equation to an equation of lower orties der, which is easier to solve. Example: Suppose we 1 1 1 need to solve the equation y 00 + p(t)y 0 + q(t)y = 0 and csc θ = , sec θ = , cot θ = sin θ cos θ tan θ we already know that y1 (t) is a solution. To find the that immediately follow from the definitions of the second solution we seek it in the form y = v(t)y1 (t) = trigonometric functions. v(t)e−t , where v is a yet to be determined function. Calculating the first and second derivatives of y and rectangular coordinate system The same as substituting back into the equation (also rememberCartesian coordinate system. ing that y1 satisfies that equation) we arrive to the recurrence formulas, relations Different rela- simple equation tions/formulas between the members (terms) of a sequence with the following common feature: To find y1 v 00 + (2y10 + py1 )v 0 = 0,

122 which is indeed an equation of the first order if we region A collection of points, a set, usually in twodenote v 0 = u. Now, this equation is always solvable or three-dimensional Euclidean space. Regions are by the method of multiplying factors. the sets where functions of two or more variables are defined. There are different types of regions dependreduction to systems of equations Any differing on certain properties. ential equation of order n > 1 could be reduced to a (1) An open region (also set) is the one that with system of n first order equations with same amount every of its points contains also a disk (circle) comof unknown functions. For example, the second orpletely inside the region. Example of an open set is 00 0 der equation y − 5y + 2y = 0 could be written as a the unit circle x1 +y 2 < 1 without the circumference. system (2) A closed region is the one that contains all of its x01 = x2 , x02 = −2x1 + 5x2 boundary points. For a plane region, for example, if we make simple substitutions x1 = y, x2 = y 0 . the point (x, y) is a boundary point if any circle with center at that point contains points that belong to reference angle For a given angle θ, −∞ < θ < ∞ the region and points that do not belong to it. An in standard position, the reference angle (usually deexample of the closed region is the ”closed” unit circle 0 noted by θ ) is the acute angle formed by the terminal 2 2 x + y ≤ 1. side of the given angle and the horizontal axis. Hence, if the terminal side is in the first quadrant, then the (3)Connected region is the one where any two points reference angle is equal to the given angle (possibly of that region could be connected by a curve entirely subtracted or added some multiple of 360◦ to make it belonging to the region. acute). Using degree measure, here are the formulas (4)Simply connected region is the one where every to calculate the reference angles in cases when the simple, connected curve encloses only points that are in the region. Roughly speaking, simply connected given angle is between 0◦ and 360◦ . regions do not have any ”holes” or more than one ◦ 0 1) If 0 < θ < 90 , then θ = θ; ”piece”. ◦ ◦ 0 ◦ 2) If 90 < θ < 180 , then θ = 180 − θ; 3) If 180◦ < θ < 270◦ , then θ0 = θ − 180◦ ; 4) If 270◦ < θ < 360◦ , then θ0 = 360◦ − θ. reflection of a function A function (and its graph) could be reflected with respect to any line of the plane. For a given function f (x) the function g(x) = f (−x) is its reflection with respect to the yaxis, and the function h(x) = −f (x) is its reflection with respect to the x-axis. The reflection of the graph of the function f (x) with respect to the line y = x represents the graph of the inverse function f −1 (x), if the inverse exists. reflection transformation Or reflection operator, is a transformation that maps each vector into its symmetric image about some line or plane. Example: The operator T : R2 → R2 that transforms any vector to its symmetric with respect to the y-axis is given by the standard matrix −1 0 A= . 0 1

regression line

See least squares regression line.

regular polygon A polygon that has equal sides (and also equal inner angles). Examples are the square, regular pentagon, hexagon, octagon, etc. regular singular point For series solutions of linear differential equations of the type P (x)y 00 + Q(x)y 0 + R(x)y = 0. If for some point x0 , P (x0 ) = 0 but the limits lim (x − x0 )

x→x0

R(x) Q(x) , lim (x − x0 )2 P (x) x→x0 P (x)

are both finite, then the point x0 is a regular singular point and for the equation the series solution is still possible. See the main entry series solution. regular transition matrix A transition matrix is regular if some integer power of it has all positive entries. related rates

If two or more variable quantities

123 are related to each other then their changes are also related. If there is a formula to connect the rate of change of one variable to the rate of change of the other(s), then these rates are called related. Example: The volume of a ball (sphere) is related to the radius by the formula V = 4πr3 /3. When the radius starts increasing at the rate dr/dt, the volume also starts increasing and the relation is dV /dt = 4πr2 dr/dt.

remainder of a series Let Σan be a numeric or functional series. Then the sum of the first n terms sn = Σnk=1 ak is its nth partial sum and the infinite sum of the terms starting from n + 1 is its remainder Rn = Σ∞ n+1 ak . In case the series is convergent, Rn = s − sn .

remainder estimates for series For most of the numeric or functional series it is either very difficult or even impossible to find the exact value that’s why relative error If in some calculation an error is we try to find their approximate values. The best way made then the difference between the exact value and to do it is to take some partial sums of that series. the approximate value is the absolute error. The ra- In that case the part we ”cut” from the series (the tio of the absolute error and the the exact value is infinitely many terms from the point we stop) is the the relative error. remainder of that series. To estimate the accuracy of the approximation it is essential to have good estirelative growth rate The growth rate is the meamates for these remainders. Below are two examples sure of the change of a variable quantity in a unit of estimates of remainders: time, or its instantaneous change. If we divide that (1) Suppose f (k) = ak , where f is a continuous, posrate by the quantity itself, the result will be the relitive, P and decreasing function for x ≥ n and suppose ative growth rate. For example, if P denotes the size that an is convergent. If Rn is the remainder of of some population, then dP/dt is its growth rate and that series, then 1 P dP/dt is the relative growth rate. Z ∞ Z ∞ relative maximum and minimum The same as f (x)dx ≤ Rn ≤ f (x)dx. n+1 n local maximum and minimum. P remainder (1) If an integer (dividend ) is divided (2) Let s = (−1)n−1 bn and sn is its partial sum. by another (divisor ) and the result is not an integer, Assume also that 0 ≤ bn+1 ≤ bn and limn→∞ bn = 0. then whatever is left over is the remainder. For ex- Then |Rn | = |s − sn | ≤ bn . ample, when we divide 127 by 5, the result is 25 and For the estimates for remainders of power series see the remainder is 2. In cases when division is possible Taylor series. without remainder, we say that the remainder is zero. removable discontinuity For functions that are (2) If a polynomial is divided by another polynomial, not continuous (have discontinuity) at some point the result is some third polynomial and whatever is x = c, it is sometimes possible to redefine them at left over is the remainder of the division. In the divithat point so they become continuous. This kind sion of discontinuity is called removable. Examples: The 7x − 13 2x3 − 3x2 + x − 4 function f (x) = (x2 − 1)/(x − 1) is not defined at = (2x − 3) + 2 2 x +3 x +3 the point x = 1 (because the denominator is equal to the polynomial 7x − 13 is the remainder. The degree zero there). Now, we can redefine this function as of the remainder is always lower than the degree of 2 x −1 the divisor. if x 6= 1 f (x) = x−1 2 if x = 1 remainder theorem If a polynomial f (x) is divided by a binomial x − k, then the remainder is and this function will be continuous everywhere. In r = f (k). the case of the function f (x) = 1/x, on the other As a result, it follows, that if the polynomial is divis- hand, it is impossible to redefine the function so it ible by the binomial, then the remainder is zero. becomes continuous at the point x = 0, hence the

124 discontinuity at that point is not removable. representation of functions Functions could be represented in many different ways, the most common being the verbal, graphical, as a list or table, and algebraic or analytic. For specific examples see the entry functions.

by one variable and then by the second one or do integration in reverse order (with appropriate change of limits of integration). Exact conditions when this is possible are given by Fubini’s theorem. Example: Z 0

representation of vector Any vector in a vector space V could be represented as a linear combination of vectors of any basis of the space V . See corresponding entries for more details. residual The least squares regression line gives the predicted values of the variable in the given interval that are not necessarily equal to the observed values of the variable. The difference between the predicted value and the observed (actual) value of the variable is the residual.

1

Z

1

sin(y 2 )dydx =

x

Z 0

Z = 0

1

y sin(y 2 )dy =

1

Z

y

sin(y 2 )dxdy

0

1 (1 − cos1). 2

rhombus The special case of parallelogram, when all four sides are equal.

Riemann integral Let f (x) be a continuous function on the closed finite interval [a, b]. Let us divide the interval into n equal parts of size ∆x = (b − a)/n. Denote the endpoints of these parts by x0 (= a), x1 , x2 , · · · , xn (= b) and chose arbitrary resonance Many natural systems (mechanical, points x∗1 , x∗2 , · · · , x∗n in each of these smaller intervals electromagnetic, acoustic, etc.) have their specific [xi−1 , xi ], 1 ≤ i ≤ n. Then the Riemann integral of vibration frequencies. If an outside force, acting f on the interval [a, b] is the limit periodically, is applied to the system, and its own Z b n X frequency is close or identical to the system’s fref (x)dx = lim f (x∗i )∆x. quency, then the phenomenon of resonance occurs. n→∞ a i=1 In that case the amplitude of vibration of the system increases significantly even with very small outEquivalently, the integral can be defined with the side force. Mathematically this could be expressed sample points x∗i chosen to be the left endpoints, right by equation of the form endpoints or the midpoints of intervals. 00 0 The sum in the definition of the integral is called a my + γy + ku = F0 cos ωt, Riemann sum. See also definite integral. where the right side represents the outside periodic right angle An angle that measures 90◦ in degree force. measure or π/2 in radian measure. response variable In Statistics, another name for right triangle Any triangle that has one right andependent or y-variable. See also explanatory varigle. The other two angles in such a triangle are necable. essarily acute. revenue function A function (in business and ecoright circular cylinder A cylinder that has a cirnomics) that represents the revenue from production cle as its base and the sides are perpendicular to the (or sales) of x units. If p(x) is the price (or demand) base. function, then the revenue function is given by the right-hand derivative If a function is not differformula R(x) = xp(x). entiable at some point x = c, then the limit in the reversing order of integration When evaluating definition of derivative does not exists, i.e. double integrals (and, more generally, multiple integrals) it is important to be able to reverse the order of f (c + h) − f (c) lim integration. This means that one can integrate first h→0 h

125 does not exist. In some cases, however, one sided limit might exist. If the right-hand limit

that make sense under the definition above: √ √ √ √ √ n n 1) n am = ( n a)m , 2) n a · b = ab, f (c + h) − f (c) r √ q n lim √ a a m √ n n h h→0+ 3) √ = , 4) a = nm a, n b b √ exists, then it is called right-hand derivative of f at 5)( n a)n = a, the point x = c. See also left-hand derivative. √ √ 6) n an = |a| for n even; n an = a for n odd. right-hand limit Let f (x) be a function defined on some interval [a, b] (also could be an open inter- The roots are used√also to define fractional exponents, m val ). Right-hand limit of f at some point c is the because an/m = an and they obey the same rules limit of f (x) as the point x approaches c from the (properties) as the n-th roots. right, or, which is the same, as x > c. The notation (2) When solving algebraic equations, the term ”root is limx→c+ f (x). The precise definition of right-hand of the equation” is used as a synonym of the terms limit (the ε − δ definition) is the following: The ”solution of the equation” or ”zero of the equation”. function f has right-hand limit at the point c and See zeros of the equation. √ that limit is L, if for any real number ε > 0 there root function The function f (x) = n x, where n exists δ > 0 such that |f (x) − L| < ε as soon as is a positive integer. If n is even, then the domain |x − c| < δ, x > c. See also limit and left-hand limit. of this function is [0, ∞) and for odd n the domain of all real Roll’s theorem This theorem states that if a is the set √ √ numbers. The graphs show the differentiable function has the same values at the functions x and 3 x respectively. endpoints of some closed interval, then somewhere inside that interval the tangent line to this function should be horizontal. Formally: Let f be continuous on some interval [a, b] and differentiable inside the interval (a, b). If, additionally, f (a) = f (b), then there is a number c, a < c < b, such that f 0 (c) = 0.

root (1) Has the same meaning as the radical. Let n ≥ 2 be an integer and a and b– some real √ numbers such that b = an . Then we write a = n b and call a the n-th root of b. Not all real numbers have real n-th roots. For example, the number −3 has no real square root because the square of any real number is positive (or zero). If a real number has a real root, then the one that has the same sign as the number is called the principal n-th root. To guarantee the existence of the roots, the following definitions are given: (a) If n is even and a ≥ 0, then the n-th root of a is the positive number b such that bn = a. For the case of negative number a see roots of a complex number ; (b) If n is even and a is any real number, then the n-th root of a is the real number b such that bn = a. The following properties are valid for all radicals

roots of a complex number Any complex number z has exacly n roots of order n (denoted by wk , k = 0, 1, 2, · · · , n − 1). To find the nth root of a complex number z = a + ib, we write it in trigonometric form z = r(cos θ + i sin θ) and use the formula √ θ + 2πk θ + 2πk wk = n r cos + i sin , n n

126 √ where n r means the positive nth root of the positive number r. Example: To √ find the four fourth roots of the number z = −2 + 2 3i we write z = 4(cos 120◦ + i sin 120◦ ) and wk , k = 0, 1, 2, 3 are given by √ 120◦ + 360◦ k 120◦ + 360◦ k 4 4 cos + i sin , 4 4

rotations about the origin to the angle θ is given by the matrix cos θ − sin θ . sin θ cos θ

There are more rotation operators in three dimensional Euclidean space, corresponding to rotation about the origin or about some axis. for example, the counterclockwise rotation about the positive x√ √ √ axis through an angle θ is given by the matrix from where we find w0 = 26 + i 22 , w1 = − 22 + √ √ √ √ √ i 26 , w2 = − 26 − i 22 , w3 = 22 − i 26 . 1 0 0 0 cos θ − sin θ . roots of unity The nth roots of the number 1 con0 sin θ cos θ sidered as a complex number. We can write the number 1 in the trigonometric form as 1 = cos 0 + i sin 0 Similar matrices work for rotations about y-axis or and apply the formula for the roots of a complex num- z-axis. ber and get round-off error Also called rounding error. The difference between the actual (exact) value and the 2πk 2πk wk = cos + i sin , k = 0, 1, · · · , n − 1. numeric approximation value. Round-off errors are n n inevitable when representing irrational (and some The first root of unity is always 1(w0 = 1). Geomet- rational) numbers because their numeric value conrically, the roots of unity are points located on the tains infinitely many digits (too many digits, respecunit circle at the equal distance from each other. tively). √ Example: The exact decimal value of the 2 contains infinitely many digits (and, as number root test A method of determining the absolute a result, will never be known). Depending on situconvergence or divergence of a (numeric or funcation we may use the numeric approximations 1.41, tional) series. p n 1.4142, 1.414213562, etc. The round-off errors for (a) If limn→∞ |an | = a < 1, then the series P ∞ these values are approximately 0.0042, 0.00001356, convergent. n=1 an is absolutely p 0.000000000373. n (b) If limn→∞ |an | = a > 1, then the series is divergent. row-echelon form A matrix is in the row-echelon (c) In the case a = 1 the test cannot give a definite form if the following conditions are satisfied: answer. (1) If a row does not consist of zeros only, then the root law for limits If n is a positive integer, then first non-zero element is 1 (leading 1); (2) Rows consisting of zeros only are grouped at the q p n bottom; n lim f (x) = lim f (x). x→a x→c (3) In two non-zero rows, leading 1 in the bottom row is to the right of the leading 1 of the row above. In case when n is even, we need additionally assume The matrices that limx→c f (x)√> 0. √ In the special case f (x) = x n n we have limx→c x = a. 1 −2 0 1 3 0 −2 0 1 0 0 0 1 3 . rotation operator Also called rotation transfor0 0 1 0 0 0 0 mation. The operator in the Euclidean space that rotates each vector of that space with respect to some are in the row-echelon form. fixed point or some fixed axis by some specific angle. The rotation operator in the plane corresponding to row equivalence If a matrix A can be obtained

127 from another matrix B by a finite sequence of elementary row operations, then matrix B can be obtained from matrix A by the same number of elementary row operation. Because every elementary row operation is invertible, the second sequence of matrices is just the inverses of the first sequence (in reverse order). These kind of matrices are called row equivalent.

S

saddle point Suppose the point (a, b) is a critical point for a function f (x, y). If it is neither a maximum nor minimum value for the function, then the corresponding point on the graph of f (a point on the row reduction The process of applying the surface determined by the function) is called a saddle Gauss or Gauss-Jordan elimination procedure to a point. given matrix. This process is useful, in particular, in sample A part of the population gathered with the evaluating the determinants of matrices. intention of getting information about the popularow space For an m × n matrix A the vectors tion. The process of gathering samples is called samformed by the m rows of that matrix are called row pling. In order to present statistical (and scientific) vectors and the subspace of the Euclidean space Rn value all samples should be random. See the entry spanned by that vectors is the row space of the simple random sample. In addition to simple ranmatrix. The row spaces of row equivalent matrices dom sample we have the following special methods of sampling: are the same. 1) Probability samples give each element of the popruled surface A surface that could be generated ulation equal chance to be chosen; by the motion of a straight line. Examples of ruled 2) In stratified random sample we first divide popsurfaces are cylinders, cones, hyperbolic paraboloids. ulation into groups of similar objects (stratas) and then used the simple random sample in each of these groups. The results should be put together; 3) In systematic samples we chose every nth element of the population. There are some other methods of sampling all of which assure the un-biased character of the sample. sample mean The mean (average) of all sample values. If x1 , x2 , · · · , xn are the sample values then the mean is Pn xi x1 + x2 + · · · + xn x= = i=1 . n n sample space When considering a random phenomena, the collection of all possible outcomes is the sample space. For example, when rolling a dye, the sample space is the collection of the outcomes 1, 2, 3, 4 ,5, and 6. sample standard deviation The square root of √ the sample variance: s = s2 . See also standard deviation. sample variance If x1 , x2 , · · · , xn are all the values of the sample and x is the sample mean, then the

128 variance is given by the formula Pn (xi − x)2 s2 = i=1 . n−1 There is another formula, called shorthand formula, that is sometimes more convenient to use: Pn Pn 2 n i=1 x2i − ( i=1 xi ) 2 s = . n(n − 1) See also variance. sampling distribution Suppose we have a distribution from large population. Chose a simple random sample of size n from that population. After that we find the mean of that sample and denote it by x ¯1 . Then we do the same process the second time and denote the mean of the second sample of size n by x ¯2 . If we repeat this process many times we will get a sequence, or set of the values {¯ x1 , x ¯2 , x ¯3 , · · ·}. The distribution formed from these values is the sampling distribution. According to the Central limit theorem, the sampling distribution is always normal (more precisely, approaches to normal, if n increases). The same theorem also establishes relationship between the means and standard deviations of these distributions.

scalar product Has the same meaning as the inner product. Let x, y be two vectors. Then (1) hx, yi = hy, xi; (2) For any real a, hax, yi = ahx, yi; (3) hx, xi ≥ 0 and hx, xi = 0 if and only if x = 0. scalar multiplication The operation of multiplying vectors or matrices by a scalar. In the case of matrices, to multiply by a scalar means to multiply each entry of the matrix by the same constant. Hence, 1 0 −2 2 0 −4 2 · −1 3 4 = −2 6 8 . 5 −2 0 10 −4 0 In the case when vectors are given in the column or row form, then scalar multiplication works exactly as in the case of matrices. If the vectors are given in an abstract vector space, then scalar multiplication is defined in the form of axioms (part of the axioms of a vector space): (1) If k is a scalar and u is in the space V , then ku also is in V ; (2) k(u + v) = ku + kv; (3) (k + l)u = ku + lu; (4) k(lu) = (kl)u.

scatterplot Also called scatter diagram or scatsawtooth wave The function defined as f (t) = ter graph. If the statistical data comes in pairs t, 0 ≤ t < 1 and then repeated periodically with (paired data), then they could be viewed as ordered pairs and presented on the Cartesean plane as the period 1: f (t + 1) = f (t). a number of points. The result is called scatterscalar The same as a constant. A quantity that plot. The figure shows the scatterplot of paired does not change. Real and complex numbers are data {(2, 24), (3, 37), (5, 45), (8, 31), (9, 78), (11, 61), scalars. (12, 82)}. scalar equation of a plane The same as equation of the plane and in three dimensional space is given by ax + by + cz + d = 0. scalar field Or the field of scalars. In choosing the scalars (constants) to use in different situations, we usually choose among the fields of rational numbers, real numbers or complex numbers. When we need a real vector space, then the choice is the field of real numbers. Also we consider polynomials using coeffi- scientific notation Notation used to write very cients from the field of rational numbers. large or very small numbers in a more compact form.

129 In this form the number is written as a number between 1 and 10 multiplied by some power of 10. Examples: (1) The number 4250000000 can be written as 4.25 × 109 . (2) The number 0.0000716 could be written as 7.16 × 10−5 . secant function One of the six trigonometric functions. Geometrically, the secant of an angle in a right triangle, is the ratio of the hypothenuse of the triangle to the adjacent side. Also could be defined as the reciprocal of the cosine function. The function sec x could be extended to all real values exactly as the cos x function. The domain of sec x is all real values, except x = π/2 + πn, n any integer, and the range is (−∞, −1] ∪ [1, ∞). sec x is 2π-periodic. One of the Pythagorean identities relates the secant function to the tangent function: 1 + tan2 θ = sec2 θ. The following are the calculus related formulas: d (sec x) = sec x tan x, dx Z sec xdx = ln | sec x + tan x| + C.

sector of a circle A part of a circle that is bounded by two radiuses and the arc of the circumference connecting the endpoints of radiuses. semi-circle One half of a circle. Also could be viewed as a sector formed by two radiuses making 180◦ angle. separable differential equations In the most general setting, the equation is separable if it could be written in the form M (x) + N (y)

dy = 0. dx

Another way of describing separable equations is to write them in the form dy f (x) = . dx g(y) To solve this equation we write g(y)dy = f (x)dx and integrate to find the general solution Z Z g(y)dy = f (x)dx. Example: Solve the equation dy/dx = x3 /y 2 . Solution: y 2 dy = x3 dx, Z Z 2 y dy = x3 dx, y 3 /3 = x4 /4 + C, and y = p 3 3x4 /4 + K.

p 3 3x4 /4 + 3C =

sequence A sequence is a list of numbers, funcsecant line A line that connects two points on tions, vectors, etc. written in some order. Terms (or some curve. The segment of the secant line between members) of a sequence are numbered using the set of natural or whole numbers or all integers. these two points is the chord. (1) Numeric sequences. A sequence could be viewed second derivative test A test that allows to find as a function of natural (whole, integer) variable. local maximums or minimums and substitutes the Hence, we write f (1) = a1 , f (2) = a2 , · · ·. A sequence first derivative test in cases when the second deriva- is finite if it has only finite number of terms. Othertive at the critical points exists. wise it is called infinite. A sequence {an } is bounded, Assume f 00 (x) is continuous near the point c. if there is a number M > 0 such that |an | ≤ M for (a) If f 0 (c) = 0 and f 00 (c) > 0, then f has a local all n. {an } is called increasing if any next term is minimum at the point c; greater than the previous one: an+1 > an . Similarly, (b) If f 0 (c) = 0 and f 00 (c) < 0, then f has a local it is called decreasing if an+1 < an . The general name maximum at the point c. for increasing and decreasing sequences is monotonic.

130 Limits for sequences are defined like limits for functions. A sequence {an } is said to have a limit L as n → ∞ if for any > 0 there is a N > 0 such that |an − L| < as soon as n > N . We write limn→∞ an = L. A sequence that has a limit as n → ∞ is called convergent. Otherwise it is divergent. In case when the sequence is divergent in a specific way that it increases without bound, we say that it goes to infinity. Formally, limn→∞ an = ∞ means that for any M > 0 there is an integer N such that an > M as n > N . The case limn→∞ an = −∞ is defined similarly. (2) Sequences of functions could be treated similar to numeric sequence. Again, we have the notions of bounded , increasing or decreasing sequences. The most common functional sequences are sequences of monomials {xn } and trigonometric functions {sin nx, cos nx}. series The sum of the terms of a finite or infinite, numeric or functional sequence. Accordingly, we have numeric or functional series.To write the series we use the summation (sigma) notation. P∞ 1) Numeric series. The series n=0 an is called convergent, if the sequence of its partial sums Sn = Pn a is a convergent sequence. k=0 k P∞ The precise definition is: We say that the series n=0 an is convergent and converges to some finite number S, if for any given > 0 there exists a natural number N such that |Sn − S| < as soon as n ≥ N . In order to find out if a series is convergent or not, various convergence tests may be used. For positive series see integral test, comparison test, lomit comparison test. For alternating series see the alternating series test. Finally, for absolute convergence of the series see the ratio test or the root test. 2) Functional series. If the terms of the series consists of functions, then it is called functional series. A functional series may be convergent for some values of the variable and divergent for other values. The definition of convergence of a functional series at any point is exactly the same as the definition of the convergence of a numeric series. The most common functional series are power series and trigonometric series. Besides them, other functional series are also considered (such as Bessel series or series of Laguerre

polynomials). See also binomial series, Taylor series, MacLaurin series. series solution For linear differential equations. Assume that we are solving the equation y (n) + a1 (x)y (n−1) + · · · + an (x)y = 0, where the coefficients a1 (x), a2 (x), · · · , am (x) are analytic functions, hence can be represented by their Taylor-MacLaurin series. The idea behand the series solutions of this equation is that it is natural to look for analytic solutions, i.e. solutions, represented by power series. This idea extends also to some cases when the coefficients are not analytic but have some ”regularity” properties. The method works the best when the coefficients ai (x) are polynomials, or even better, if they are monomials. We will demonstrate this method for the case of second order equation P (x)y 00 + Q(x)y 0 + R(x)y = 0.

(1)

A point x = x0 is called ordinary point for this equation, if P (x0 ) 6= 0, otherwise it is called a singular point. (A) The case of the ordinary point. Dividing the equation by P (x), we will have a simpler form near the ordinary point x0 (because P (x0 ) 6= 0) y 00 + p(x)y 0 + q(x)y = 0

(2)

and will look for a solution y(x) =

∞ X

an (x − x0 )n ,

n=0

where the coefficients an are still to be determined. If we calculate the first and second derivatives of this unknown function and also write power series representations of functions p(x) and q(x) around the point x0 , then we will get ∞ X

n(n − 1)an (x − x0 )n−2

n=2

+

∞ X j=0

j

pj (x − x0 )

∞ X n=1

! nan (x − x0 )

n−1

131 ∞ X

!

∞ X

!

(B) The case of the singular point. If the function P (x) from the equation (1) above becomes zero at n=0 i=0 any point, then the described method does not work. Finally, if we perform all multiplications and addi- There is however an exceptional case where a similar tions, equate resulting coefficients of all powers to method gives a satisfactory solution. If P (x0 ) = 0 zero (because on the right side we have the identi- and cally zero function), then we will find expressions for Q(x) R(x) coefficients an we the help of the known coefficients , lim (x − x0 )2 lim (x − x0 ) x→x0 x→x0 P (x) P (x) pj , qi of the functions p(x) and q(x). Example: Find the series solution of Airy’s equation are both finite, then the point x0 is called regular singular point. Now assuming for simplicity that the y 00 − xy = 0, −∞ < x < ∞. point x = 0 is a regular singular point for the equation (1) we divide that equation by P (x) and then Proceeding as described before and using the fact multiplying everything by x2 get the simplified equathat the functions p(x) = 0 and q(x) = x are very tion simple, we will have series solution presentation centered at x = 0 (here the summation index is changed x2 y 00 + x[xp(x)]y 0 + [x2 q(x)]y = 0. (3) to a more convenient form): Here we are seeking solutions in the form ∞ ∞ X X ∞ X 2a2 + (n + 2)(n + 1)an+2 xn = an−1 xn . y = xr an xn , +

qi (x − x0 )

n=1

i

n

an (x − x0 )

= 0.

n=1

n=0

Now, clearly a2 = 0, because the right side does not have a constant term and from the recurrence relation

where both the coefficients an and the index r are to be determined. Proceeding as in the case of ordinary points and equating the coefficients of equal (n + 2)(n + 1)an+2 = an−1 powers we will have a recurrence relationship for the that follows from equating the coefficients of equal coefficients and the index: n−1 powers, we have also a5 = a8 = · · · = 0, or a3n−1 = X 0, n = 1, 2, 3, · · ·. Similarly, we find that F (r + n)an + ak [(r + k)pn−k + qn−k ] = 0, a3n = and a3n+1

a0 2 · 3 · 5 · 6 · · · (3n − 1)(3n)

a1 = 3 · 4 · 6 · 7 · · · (3n)(3n + 1)

and the solution of the equation (2)is given by the formula x3 x6 y = a0 1 + + + ··· 2·3 2·3·5·6 x4 x7 +a1 x + + + ··· . 3·4 3·4·6·7

k=0

where F (r) = r(r − 1) + p0 + q0 . Theorem. (a) Let r1 ≥ r2 be the two real roots of the indicial equation F (r) = 0. Then there exists a solution of the form " # ∞ X r1 n y1 = |x| 1+ an (r1 )x , n=1

where an (r1 ) are found from recurrence relation above. (b) If r1 − r2 is not zero or a positive integer, then there exists a second linearly independent solution of the form " # This solution depends on constants a0 and a1 . If we ∞ X need to find a unique solution then it is necessary to y2 = |x|r2 1 + an (r2 )xn . have two initial or boundary values given. n=1

132 (c) If r1 = r2 , then the second solution is given by y2 = y1 (x) ln |x| + |x|r1

∞ X

conic. See the equations in the corresponding entries.

bn (r1 )xn .

shift of a function The process of moving the graph of a function horizontally or vertically or both. Algebraically the vertical shift is achieved by adding (d) If r1 − r2 = N is a positive integer, then or subtracting a number from the function: g(x) = " # f (x) + c is the vertical shift if the functionf . If c > 0 ∞ X r2 n then the function is shifted up and if c < 0 then the y2 = ay1 (x) ln |x| + |x| 1+ cn (r2 )x . shift is down. The horizontal shift of the function f n=1 is given by the formula g(x) = f (x − c). If c > 0 then The coefficients an (r1 ), bn (r1 ), cn (r2 ) and the con- the function shifts to the right and if c < 0 it shifts stant a can be determined by substituting the solu- to the left. For example, the graph of the function tions into the equation (3). The constant may equal g(x) = (x + 3)2 + 2 could be found from the graph of zero and the corresponding term will be missing. the parabola f (x) = x2 by shifting it left 3 units and sets One of the most basic notions of mathematics. shifting up 2 units. By a set we understand any collection of objects of sieve of Erathosthenes The ancient method of any nature: numbers, people, countries, stars, etc. finding prime numbers. Suppose we need to find all Objects of a set are called its elements or members. the prime numbers up to some number N . Make the Two sets A and B are considered equal (more pre- list (table) of all these numbers and start eliminatcisely, identical) if they have exactly the same ele- ing the numbers that are not prime as follows: First ments. A set B belongs to another set A (is a subset throw away the number 1 because it is not prime. of A, notation: B ⊂ A), if all elements of B are Next, leave 2 because it is prime but throw away all also elements of A, but the opposite is not necessar- multiples of 2 (all even numbers) up to N . In the ily true. For a set A, its complement A is the set of third step leave 3 but eliminate all the multiples of all elements that do not belong to A. For example, in 3 that are still in the list (all even numbers that are the larger set of all real numbers, the complement of multiples of 3 were eliminated in the previous step). all rational numbers is the set of all irrational num- Continuing this way we will throw away all non-prime bers. For the operations of intersection and union of numbers and what is left would be all prime numbers sets see corresponding entries. up to N . During this process our number list gets lots shift of index of summation In summation no- of ”holes” which is the reason for the name ”sieve”. n=1

tation for functions it is sometimes important or con- sigma notation The same as summation notation. venient to ”shift” the index of summation to be able to combine two or more sums. For example, the sum signed elementary product An elementary product multiplied by +1 or -1. Used to calculate detern X minants. ak−1 xk signed number A term that is sometimes used to k=1 indicate negative number, as numbers preceded by Pn−1 could be written also as k=0 ak xk+1 using the sub- the ”minus” sign. stitution m = k − 1 and then returning to the index significance level Also called alpha level and dek because the notation does not make the sum any noted by the Greek letter α. For tests of significance, different. a numeric level such that any event with probability shifted conics The three major conic sections (el- below that is considered rare. It is set arbitrarily but lipses, parabolas, hyperbolas) may be shifted horizon- the usual choices are 0.1, 0.05 and 0.001. The choice tally or vertically or both. The result is a shifted of the significance level depends on the importance

133 of the problem. significance tests

See test of significance.

similar terms The same as like terms. similar matrices Square matrices A and B are similar if there exists an invertible matrix P such that B = P −1 AP . similar matrices have many properties and characteristics in common. In particular, the determinants, traces, eigenvalues, nullity of similar matrices are the same. If the two matrices are the standard matrices of some transformations T1 and T2 then the transformation bringing one to the other is called similarity transformation simple curve A curve is simple if it does not intersect itself, except maybe the endpoints. If the curve is given by the parametric equations x = f (t), y = g(t), a ≤ t ≤ b, then being simple means (f (c), g(c)) 6= (f (d), g(d)) for c, d 6= a and c, d 6= b. If the endpoints coincide, i.e., if f (a) = f (b) and g(a) = g(b), then we call it a simple closed curve. simple region A term sometimes used to describe regions that lie between the graphs of two functions. Similarly, simple solid region is a region that lies between two surfaces described by functions of two variables. simple eigenvalues See eigenvalue of matrix. simple harmonic motion If a particle or an object moves (oscillates) according to the equation y = A cos(ωt − δ) then this kind of motion is simple harmonic. Here A is the amplitude, ω/2π is the frequency and δ is the phase shift.

ple from the population where every possible triple of values from P has equal chance to be chosen. More generally, a simple random sample of size n is a sample where every possible set of n values from P has exactly the same chance to be chosen. simply connected region If any simple closed curve in the region contains only points from the region, then it is called simply connected. Another, simpler definition says, that if any two points in the region can be connected by a continuous curve that is completely inside the region, then it is simply connected. In other words, the region basically consists of one ”piece”. Simpson’s rule One of the methods of approximate integration. To calculate the approximate value Rb of the integral a f (x)dx using Simpson’s rule we divide the interval [a, b] into even number n equal intervals of the length ∆x = (b − a)/n with the points x0 = a, x1 = a + ∆x, x2 = a + 2∆x, · · · , xn = b. Then we have the approximation Z

b

f (x)dx ≈ a

∆x [f (x0 ) + 4f (x1 ) + 2f (x2 ) 3

+4f (x3 ) + · · · + 2f (xn−2 ) + 4f (xn−1 ) + f (xn )]. For the degree of accuracy of this approximation see error estimate for Simpson’s rule.

sine function One of the six trigonometric functions. Geometrically, the sine of an angle in simple interest If the bank pays interest on the a right triangle is the ratio of the opposite side principal amount only then the interest is called sim- to the hypotenuse of the triangle. More general ple. Mathematically, if P is the amount of the money approach allows to define sin x function for any real invested at the rate r yearly, then after n years that x: Let P = (a, b) be any point on the plane other principle will become P + nrP = P (1 + nr). See also than the origin and θ is the angle formed by the positive half of the x-axis and the terminal side, compound interest. connecting the origin and P . Then sin θ = √a2b+b2 . simple random sample One of the most imporNext, after establishing one-to-one correspondence tant notions in statistics. A simple random sample between angles and real numbers, we can have the of size 2 from a given population P is a sample where sine function defined for all real numbers. The range every possible pair of values has equal chance to be of sin x is [−1, 1] and it is 2π-periodic. chosen. A simple random sample of size 3 is a sam-

134 and 5 columns then we say that it is of size 4 × 5. skew lines Two lines in three dimensional space are called skew lines if they are not parallel and do not intersect. Skew lines do not lie in any plane. skew-symmetric matrix A square matrix is skew-symmetric if it is equal to the opposite of its transpose: A = −AT . The sine function is related to other trigonometric functions by various identities. The most important is the Pythagorean identity sin2 x + cos2 x = 1. The derivative and indefinite integral of this function are: Z d (sin x) = cos x, sin xdx = − cos x + C. dx sine integral function The function defined by the integral Z x sin t dt. Si(x) = t 0

slant asymptote An asymptote that is neither horizontal nor vertical. For exact definition see the entry asymptote. slope One of the most important characteristics of a line on a plane. Shows the level of ”steepness” or ”flatness” of the line. If the points (x1 , y1 ) and (x2 , y2 ) are on the line, then, by definition, the slope of the line is the number m = (y2 −y1 )/(x2 −x1 ). For a horizontal line the slope is zero and for a vertical line the slope is not defined because the denominator in the definition becomes zero. slope field

Same as direction field.

The integrand cannot be expressed with elementary slope-intercept equation of a line If the slope function and the values of this function are found by m and the y-intercept b of the line are known, then the equation of the line is given by the formula approximate (numeric) integration. singular matrix The opposite of an invertible matrix. For a square matrix A to be singular means there is no matrix B such that A · B = I, where I is the identity matrix. An easy criteria for the singularity is that the determinant of a singular matrix is zero.

y = mx + b. Example: If the slope of the line is m = 2 and the y-intercept is the point (0, −1), then the equation of the line is y = 2x − 1. See also line. smooth curve A curve created by a smooth function or smooth functions.

singular point A point where the function cannot be defined because it has infinite limit (from the left, smooth function Usually, a function that has confrom the right, or both). The point x = 1 is singular tinuous derivative. In some other settings smooth function might mean a function that has more than for the function f (x) = 1/(x − 1)2 because one derivative. In particular, infinitely differentiable 1 functions are also called smooth. The notion of the lim = ∞. x→1 (x − 1)2 smooth function extends also to functions of several variables. A function has singular point at infinity if limx→∞ f (x) = ±∞. For the use of the term singular smooth surface A surface given by a differentiable in differential equations see irregular singular point, function f (x, y) of two variables. Geometrically this regular singular point. means that the surface has no ”corners” or ”edges”. size of a matrix Size of a matrix is the number of rows and columns it has. So, if the matrix has 4 rows

solid Usually, a three dimensional object, geometric figure. For example, solid angle means the figure

135 that we receive when we connect a fixed point with all the points of a closed simple curve in space. Solid of revolution means a solid that is received by rotating a plane region about some line, such as coordinate axis. solution Depending on the type of equation the solution would be a number, set of numbers, set of ordered pairs or ordered triples, a matrix, a vector, a function or set of functions, etc. All these solutions have the same common property: substitution into the equation (or system of equations) results in a true statement, an identity. For details of how to find solutions of equations see the articles quadratic equation, cubic equation, quartic equation, rational equations, radical equations, systems of linear equations, linear ordinary differential equation and many others where the solutions of the corresponding equations are presented. solution curve The graph of the solution of a differential equation. solution set The set of all solutions of the given equation.

and the sides are denoted by a, b, c and we follow the convention that side a is opposite to angle A, side b is opposite to angle B and side c is opposite to angle C. (1) The case AAS. Let C = 102.3◦ , B = 28.7◦ , b = 27.4. First we find the angle A using the fact that the sum of all angles is 180◦ : A = 49.0◦ . Then using the Law of sines we have a = b sin A/ sin B ≈ 43.06. Similarly, c = b sin C/ sin B ≈ 55.75 and the triangle is solved. (2) The case SAS. Let A = 115◦ , b = 15, c = 10. First, we use Law of cosines to find the side a : a2 = b2 + c2 − 2bc cos A ≈ 451.79 and a ≈ 21.26. After this, we have the choice of using either the Law of cosine again or the Law os sines to find one of the missing angles. Law of sines will give sin B = (b/a) sin A ≈ 0.63945 and B ≈ 39.75◦ . Now, C ≈ 25.25◦ and the triangle is solved. (3) The case SSS. Let a = 8, b = 19, c = 14. Using the Law of cosines we have, for example, a2 + b2 − c2 ≈ −0.45089 2ac from where B ≈ 116.80◦ . Now, the rest follows as in the previous case and we can find the remaining algles as before: A ≈ 22.08◦ , C ≈ 41.12◦ . The solution is complete. cos B =

solution space Let Ax = b be a system of linear equations. Each solution x of this equation is called solution vector. The set of solutions forms a linear space called solution space. space In mathematics the term has different meansolving triangles To solve a triangle means to find ings depending on context. Most commonly space one or more elements (side or angle) of the triangle means the three dimensional space we live in which knowing the other elements. To solve a triangle we is mathematically the three dimensional Euclidean need to know at least three elements (including at space. In other cases space refers to vector space. least one side), because otherwise the problem be- See corresponding entries for more details. comes impossible to solve. The follwing four cases are possible in solving triangles: (1) One side and two angles are known (AAS or ASA); (2) Two sides and one angle not formed by them is known (SSA); (3) Two sides and the angle formed by them are known (SAS), and (4) Three sides are known (SSS). The first two cases are handled by the Law of sines and the other two cases are solved by the Law of cosines. Also, the second case is called ambiguous, because it not always results in definite solution. For that case see the separate entry ambiguous case. In the following examples the angles are denoted by A, B, C

space curve A curve in three dimensional space could be given by different methods. Suppose t is a variable on some interval [a, b] and the functions f, g, h are defined and continuous on that interval. Then the set of all points (x, y, z) in space where x = f (t),

y = g(t),

z = h(t)

is a space curve given parametrically by these equations. The same curve could be also given as a vector with the equation r(t) = f (t)i + g(t)j + h(t)k,

136 where i, j, k are the standard bases in the three dimensional space. Space curve is called continuous, differentiable, etc. if the functions f, g, h have the same properties.

square matrix A matrix such that the number of rows is equal to the number of columns. There are many matrix operations that are possible for square matrices but not for other type of matrices. For example, the notions of determinant and inverse of a sphere One of the most common geometric figures. matrix do not make sense if the matrix is not square. The term has two meanings, to indicate the geometric For other entries specifically applied to square masolid or its boundary. In the first case the term ball is trices see nilponent matrix, normal matrix, skewmore commonly used. The equation of the ball (solid symmetric, symmetric, unitary matrices. sphere) of radius r centered at the points (a, b, c) is 2 2 2 2 (x − a) + (y − b) + (z − c) ≤ r . The equation of its square root The radical of the second order. The boundary (the sphere) is (x−a)2 +(y−b)2 +(z−c)2 = square root of a non-negative number a ≥ 0 is der2 . fined to be the non-negative number b ≥ 0 such that b2 = a. The √ notation for the square root of the numspherical coordinates One of the most common ber a is a. If the number a is negative then its coordinate systems in three dimensional space along square root is not a real number, it is imaginary. See with the Cartesian and cylindrical systems. It is the also the entry root for more details. three dimensional analog of the polar coordinate system on the plane. The points in spherical coordinate squeeze theorem Assume that three functions system are determined by three parameters: the dis- f (x), g(x), h(x) satisfy the inequalities f (x) ≤ g(x) ≤ tance from the origin (usually denoted by ρ) and two h(x) near some point x = c but not necessarily at that angles. The first angle, denoted by θ is measured on point. Assume also that the xy-plane from the positive x-axis and the second lim f (x) = lim h(x) = L angle, denoted by φ, is measured from the positive x→c x→c z-axis. This way ρ ≥ 0, 0 ≤ θ ≤ 2π, 0 ≤ φ ≤ π. then lim g(x) = L. x→c

Example: Evaluate the limit lim x cos(1/x).

x→0

Because the function cos(1/x) has no limit at zero, the usual product law for limits does not apply. However, since | cos(1/x)| ≤ 1, The relations between the spherical and rectangular coordinates are given by the formulas x = ρ cos θ sin φ, y = ρ sin θ sin φ, z = ρ cos φ.

−x ≤ x cos(1/x) ≤ x. Now, limx→0 x = limx→0 (−x) = 0 and by the squeeze theorem our limit is also zero.

standard basis vectors In many common vector spaces sets of vectors that form a basis are considered spiral of Archimedes See Archimedian spiral. to be standard. For example, in the Euclidean space spread of a distribution The meaning of this Rn the set of vectors e1 = (1, 0, 0 · · · , 0), e2 = term depends on situation but as a rule the spread (0, 1, 0, · · · , 0), · · · , en = (0, 0, 0, · · · , 1) is considered of any distribution is described by its standard devi- (and called) the standard basis. In the space Pn ation or variance. of all polynomials of degree n or less the set of

137 functions (1, x, x2 , · · · , xn ) is the standard basis. standard deviation For a set of values x1 , x2 , x3 , · · · , xn with the mean µ, the standard deviation is defined by the formula sP n 2 i=1 (xi − µ) σ= . n−1

am1 x1 + am2 x2 + · · · + amn xn = ym The matrix A = [aij ], 1 ≤ i ≤ m, 1 ≤ j ≤ n formed by the coefficients of these equations is called the standard matrix of the transformation T . The notion of the standard matrix could be extended also to transformations in general linear vector spaces.

standard normal distribution The special The square of this expression is called variance. case of the normal distribution, when the mean is There are other, sometimes easier formulas for calcu- equal to 0 and the standard deviation is equal to lating standard deviation. See sample standard devi- 1. If the random variable describing some normal distribution with the mean µ and standard deviation ation. σ is x then the variable z = (x − µ)/σ describes the standard equations A term used in different situ- corresponding standard normal distribution. The ations with specific meanings. For example, the stan- graph of the standard normal distribution is also the dard equation of a line is ax + by = c, or the standard normal density curve centered at 0. equation of an ellipse centered at the point (h, k) is standard position Usually refers to angles or (y − k)2 (x − h)2 vectors. In the case of angles, the standard position + = 1. is when the vertex is located at the origin and the a2 b2 initial side coincides with the positive side of the x-axis. In the case of the vectors the standard standard error In Statistics the standard deviaposition is when the initial point coincides with the tion of the sampling distribution is usually unknown. origin. See also angles, vectors. To estimate it we use values of the sample and call it sampling error because it almost never equals the standardized value The same as the z-score. actual standard deviation. The standard error for statistical inference Also called inferential the√sample mean x ¯ is given by the formula SE(¯ x) = statistics. The use of statistical information (data) s/ n, where s is the sample standard deviation and to draw conclusions about larger population. n is the size of the sample. The standard error for statistics One of the main branches of Mathematproportions is ics. It is concerned with collecting and interpreting r data. By other definitions, it is a mathematical pˆ(1 − pˆ) SE(ˆ p) = , science pertaining to the collection, analysis, intern pretation or explanation, and presentation of data. where pˆ is the sample proportion and n is the sample Some other definitions of this science also include size. the fact that Statistics most of the time deals with standard matrix Suppose T is a linear transfor- random phenomena and needs to draw conclusions mation from the Euclidean space Rn to Rm . Then from randomly received data. See different entries the vectors x = (x1 , x2 , · · · , xn ) are transferred to about Statistics throughout this Dictionary. vectors y = (y1 , y2 , · · · , ym ) by linear equations stemplot Also called stem-and-leaf plot. An easy a11 x1 + a12 x2 + · · · + a1n xn = y1 a21 x1 + a22 x2 + · · · + a2n xn = y2 ······························

and fast way of establishing the form of the distribution for relatively small data set. For example, to organize the set of values {44 46 47 49 63 64 66 68 68 72 72 75 76 81 84 88 106}, we separate the ”tens” digits and combine all the ”ones” digits that

138 correspond to each particular tens digit. The result is

important geometric figures (notions). A line is not defined but assumed to be understood as other fundamental geometric or mathematical notions. In algebra, a line is given by its equation. See the entry line for all the details. submatrix Any part of a given matrix could be considered as a matrix, called submatrix.

subset A set B is called the subset of the set A (written B ⊂ A), if any element of B is also an elestep function The function f (x) = [[x]] defined as ment of A. Example: The set {2, 4, 5} is the subset the greatest integer less than or equal to x. Differ- of the set {1, 2, 3, 4, 5}. ent variations of this function also may be called step subspace A subset W of a vector space V is called function or staircase function. a subspace if W itself is a vector space under the same Stokes’ theorem One of the most important the- rules of addition and scalar multiplication as for the orems of the multivariable Calculus. It could be space V . viewed as the generalization of the Fundamental Thesubstitution method or substitution rule. One of orem of Calculus and also of the Green’s theorem. the main methods of evaluating indefinite or definite This theorem relates surface integrals of some funcintegrals. The essence of the method is to find some tions with integrals on the boundary curve of that kind of substitution that will make the function unsurface. der the integral sign simpler and easier to evaluate Stokes’ theorem. Let S be an oriented piecewisethe integral. This method works only in cases when smooth surface bounded by a simple, closed, the function could be presented or viewed as a prodpiecewise-smooth boundary curve C with positive uct of a composite function and the derivative of the orientation. Let F a vector field with continuously function that is the argument of that function. For3 differentiable components in a region G ⊂ R that mally: contains S. Then Let u = g(x) be differentiable, then Z Z Z Z Z F · dr = curlF · dS. C S f (g(x))g 0 (x)dx = f (u)du. For definitions of vector field and curl of a vector field R√ see corresponding entries. See also divergence theo- Examples: (1) 3x − 1dx. The substitution u = rem for related results. 3x−1 results in du = 3dx and the integral transforms Z Z stretching a function One of the possible trans- to √ du 1 u = u1/2 du formations of functions. The vertical stretch of the 3 3 function f is given by the formula g(x) = cf (x), 2 2 where |c| > 1. In case when |c| < 1 the term squeez= u3/2 + C = (3x − 1)3/2 + C. 9 9 ing or compression is used. The horizontal stretching is given by the formula g(x) = f (cx) where |c| < 1. (2) R sin4 x cos xdx. Here the substitution u = In case |c| > 1 the transformation is squeezing or sin x, du = cos xdx gives compression. Z Z straight angle An angle that is 180◦ in degree measin4 x cos xdx = u4 du sure or π in radian measure. straight line

Or simply a line. One of the most

= u5 /5 + C = sin5 x/5 + C.

139

u+v u−v cos , 2 2 u−v u+v Z b Z g(b) sin . cos u − cos v = −2 sin 2 2 f (g(x))g 0 (x)dx = f (u)du. a g(a) Similar formulas are also possible for other functions subtended angle An angle is subtended by an arc but hardly ever used. (usually, of a circle) if the two rays forming the angle supplementary angles Two angles that add up pass through the endpoints of the arc. to 180◦ (or π in radian measure). Two supplemensubtraction One of the four main arithmetic and tary angles form one straight angle. The version of the substitution method for definite integrals is:

algebraic operations. By subtracting two numbers we find their difference. Example: −2 − (−5) = 3. See also addition and subtraction of complex numbers, fractions, functions, vectors, matrices. subtraction formulas for trigonometric functions See addition and subtraction formulas for trigonometric functions. sum The result of addition of two or more numbers, functions, matrices, etc. Example: The sum of the functions f (x) = 2x2 +3x−1 and g(x) = x2 +5 is the function (f +g)(x) = 3x2 +3x+4. For the sum of infinitely many numbers or functions see series and power series.

cos u + cos v = 2 cos

surfaces In the simplest cases a surface is a two dimensional object in three dimensional space that could be given by some equation z = f (x, y). Depending on properties of the function f we get different type of surfaces. For example, if f (x, y) has continuous partial derivatives by both variables then the surface would be called smooth. If the function is a second degree polynomial with respect to its variables then the surface would be called quadric. The meaning of the term closed surface should be understandable by itself. For explanations of terms oriented surface, paramatric surface see the corresponding entries. For functions of three variables the equation f (x, y, z) = k for some constant k is called the level surface.

summation notation A shorthand notation for summation of many numbers or functions. Uses the surface area Suppose some surface S is given by Greek letter sigma Σ. The notation the equation z = f (x, y) and the function has con10 X tinuous partial derivatives fx , fy in some domain D. n2 Then the area of the surface S could be calculated by n=1 the formula Z Z q means that we are adding the squares of all numbers A(S) = fx2 (x, y) + fy2 (x, y) + 1 dA, from 1 to 10. The variable n is called summation inD dex or variable. In case P the index takes all natural where dA = dxdy is the element of the area of D. ∞ numbers, the notation is n=1 an . This formula is the three dimensional generalization sum rule See differentiation rules. of the arc-length formula and has similar format. sum-to-product formulas In trigonometry, the formulas u+v u−v sin u + sin v = 2 sin cos , 2 2 sin u − sin v = 2 cos

u+v 2

sin

u−v 2

,

surface integral Suppose a surface S in three dimensional space is given by the function z = g(x, y) over the domain D in xy-plane. The integral of the function f (x, y, z) over that surface (under certain regularity conditions for that surface) is defined to be Z Z f (x, y, z)dS = S

140 Z Z

q f (x, y, g(x, y)) gx2 + gy2 + 1dA. D

Here dA = dxdy is the element of the plane measure and dS is the element of the measure on the surface S and gx , gy are the partial derivatives of the function g(x, y). In particular case when f (x, y, z) = 1 we will get the surface area of S (see above). Similar formulas are valid for surfaces given by parametric that the result of the division equations and for integrals of vector functions with The bottom row means 2 is the polynomial 2x + x − 1 and the remainder is appropriate changes. For calculations of surface in−3: tegrals of vector functions with the help of volume (triple) integrals see divergence theorem. 2x3 + 3x2 − 4 3 = 2x2 + x − 1 − . x+1 x+1 surface of revolution A surface received by rotation of some curve about an axis, such as coordinate axis. Many common surfaces (hyperboloids, systems of linear algebraic equations The genparaboloids, etc.) could be viewed as surfaces of reveral system of m linear equations with n unknowns olution. is symmetric function The term may mean differa11 x1 + a12 x2 + · · · + a1n xn = b1 ent things depending on situation. The most coma21 x1 + a22 x2 + · · · + a2n xn = b2 monly symmetry with respect to one of the coordinate axis or the origin is considered. Functions sym······························ metric with respect to the y-axis are the even funcam1 x1 + am2 x2 + · · · + amn xn = bm tions and functions symmetric with respect to the origin are the odd functions. where aij , 1 ≤ i ≤ m, 1 ≤ j ≤ n and bk , 1 ≤ k ≤ n are numeric constants and x1 , x2 , · · · , xn are the unsymmetric matrices A matrix A is symmetric if knowns. For different methods of solving these kind it is equal to its transpose: A = AT . A symmetric of equations see Cramer’s rule, Gaussian elimination, matrix is necessarily a square matrix. Example: Gauss-Jordan elimination, matrix method. 1 −1 2 systems of linear differential equations (homo −1 −2 0 geneous) This type of systems are written usually 2 0 3 in the following form synthetic division A simplified algorithm of division of any polynomial by a binomial of the form x − c. This method allows to avoid writing (and repeating) the variable x in the process of division. In order to be able to use this algorithm it is necessary to write the polynomial we are dividing in the form of decreasing powers and it is also mandatory to ”fill-in” the missing powers. For example, the polynomial P (x) = 2x3 + 3x2 − 4 should be written as P (x) = 2x3 + 3x2 + 0x − 4. Now, suppose we want to divide this polynomial by x + 1. The synthetic division procedure looks as below

x01 = a11 x1 + a12 x2 + · · · + a1n xn x02 = a21 x1 + a22 x2 + · · · + a2n xn ······························ xn = am1 x1 + am2 x2 + · · · + amn xn where x1 , x2 , · · · , xn are the unknown functions. Using vector and matrix notations, this system could be written more compactly as x0 = Ax. The solution of this system depends essentially on the eigenvalues and eigenvectors of the matrix A. (1) The matrix A has n distinct real eigenvalues

141 r1 , r2 , · · · , rn with corresponding (distinct) eigenvec- r = 2 is a repeated eigenvalue and tors (ξ1 , ξ2 , · · · , ξn ). Then the general solution vector 1 of the system x0 = Ax is given by x(1) (t) = e2t −1 x(t) = c1 ξ1 er1 t + c1 ξ2 er2 t + · · · + c1 ξn ern t . is its corresponding vector solution. The second linearly independent vector solution is given by the forExample: To solve the system mula 1 1 1 0 x0 = x x(2) (t) = te2t + e2t . 4 1 −1 −1 we find the eigenvalues r1 = 3, r2 = −1 and the corresponding eigenvectors systems of linear differential equations (non homogeneous) These systems in matrix form 1 1 could be written as ξ (1) = , ξ (2) = 2 −2 x0 = ax + g(t), and the two (linearly independent solutions would be where g(t) is a vector function. The general so 1 1 (1) 3t (2) −t lution of this system can be found by adding the x (t) = e , x = e . 2 −2 general solution of the homogeneous system to any particular solution of the non-homogeneous system. (2) If the (real) matrix A has a complex eigenvalue Among many methods for finding particular solutions then it should have also its complex conjugate eigenare the methods of undetermined coefficients, variavalue: r1 = λ + iµ, r2 = λ − iµ. The corresponding tion of parameters, Laplace transform method and eigenvectors are also conjugate in the sense that if the method of diagonalization. The first three methξ (1) = a + ib then ξ (2) = a − ib. Now, the vector ods (although for equations, not for systems) are desolutions corresponding to these complex eigenvalues scribed in separate articles. Here we will describe the could be written in the following form: diaginalizathion method. λt If the matrix A is diagonalizable, then there is an inu(t) = e (a cos µt − b sin µt), vertible matrix T whose columns are the eigenvectors v(t) = eλt (a cos µt + b sin µt). of A that diagonalizes A. Denoting x = T y we will get an equation in new variable Example: The system of equations 0

x =

−1/2 −1

1 −1/2

T y0 = AT y + g(t).

x

After applying the inverse of T from the left of this equation we arrive to new equation

has eigenvalues r1 = −1/2 + i, r2 = −1/2 − i. The two linearly independent vector solutions are y0 = Dy + T −1 g(t), t t cos t sin t where D = T −1 AT is a diagonal matrix. The last , v(t) = e− 2 . u(t) = e− 2 − sin t cos t system is in fact a system of n equations where each of them contains only one variable and has the form (3) If any of the eigenvalues of the matrix A is reyk0 (t) = rk yk (t) + hk (t), 1 ≤ k ≤ n. peated then the situation is more complicated. Example: For the system Here rk are in fact the eigenvalues if the matrix A. Each of these equations is a simple first order equa1 −1 x0 = x tion and can be solved by the method of integrating 1 3

142 factors, for example. After solving it we just need to multiply the solutions by the matrix T to find the vector x and that will be a particular solution of the non-homogeneous system.

T

systems of non-linear algebraic equations There is no theory for solutions of non-linear algebraic systems but in many cases the substitution or elimination methods (but not Gaussian elimination) work here too. Example: Solve the system of equations 3x − y = 1

t-distribution Suppose the random variable x represents some distribution (normal or not) with mean µ and standard deviation σ and denote by x ¯ the means of samples of size n from that distribution. A new distribution formed by the values t=

x ¯−µ √ , s/ n

x2 − y = 5 Solving the first equation for y and substituting into the second equation results in quadratic equation x2 − 3x − 4 = 0. This equation has the solutions x = 4, −1. Substituting these values into either of the original equations gives y = 11, −4. Finally, the system has two solutions (4, 11), (−1, −4).

where s is the sample standard deviation, is called t-distribution. This distribution is generally speaking not normal even if the original distribution of the x values is normal because√of the presence of the standard error SE(¯ x) = s/ n in denominator. The t-distribution is different for different values of sample size n and it is approaching standard normal distribution as the sample size increases. The values (probabilities) corresponding to t-distribution are found from special tables, by graphing calculators, or computers. It is generally acceptable to approximate t-distributions by the standard normal distribution if n ≥ 30. The value t in the formula above is called one-sample t-statistic. tangent function One of the six trigonometric functions. Geometrically, the tangent of an angle in a right triangle, is the ratio of the opposite side of the triangle to the adjacent side. Also could be defined as the reciprocal of the cotangent function. The function tan x could be extended as a function of all real numbers with exception of x = π/2 + πn, n any integer, and the range is all of R. tan x function is π-periodic. Tangent function is related to other trigonometric functions by various identities. The most important of these are the identities tan x = sin x/ cos x, tan x = 1/ cot x and a version of the Pythagorean identity 1+tan2 x = sec2 x. The derivative and integral of this function are given by the formulas Z d 2 tan x = sec x, tan xdx = ln | sec x| + C. dx

143 if it exists and is not zero, is the tangent vector at the point r(t0 ). tautochrone This is a curve down which a particle slides freely under just the force of gravity and reaches the bottom of the curve in the same amount of time no matter where its starting point on the curve was. It turns out that the only curve satisfying this condition is the cycloid. tangential component Any vector coming out from some point of a curve, can be presented as a combination of tangent and normal vectors. The component in the direction of the tangent vector is the tangential component.

Taylor series Let the function f (x) be defined on some interval around the point x = a and has derivatives of arbitrary order at that point. Then the power series ∞ X f 0 (a) f (n) (a) (x − a)n = f (a) + (x − a) n! 1! n=0

tangent line A line is tangent to some curve at a point of that curve, if the line and curve coincide at f 00 (a) f 000 (a) + (x − a)2 + (x − a)3 + · · · that point and have no other common points in some 2! 3! neighborhood of that point. If the curve is given by is called the Taylor series of f . The Taylor series of a differentiable function f (x), then the tangent line a function may or may not converge to represent the to this curve at the point (a, f (a) has the slope function itself. The functions such that their Taylor series converges to them are called analytic. f (x) − f (a) f 0 (a) = lim A finite part of the Taylor series with summation x→a x−a from n = 0 to n = N is called the Taylor polynomial and could be given by the equation y − f (a) = of order N . The N th Taylor polynomial is usually f 0 (a)(x − a). If the function is not differentiable at denoted by T (x). If we denote the remaining part N some point then the corresponding curve does not of the Taylor series (the sum from n = N + 1) by have a tangent line at that point. RN (x), then the question of when the Taylor series tangent line method The same as the Euler converges to the function is resolved by the following method for approximate solutions of differential Theorem. If f (x) = TN (x) + RN (x) and equations. lim RN (x) = 0 N →∞ tangent plane A plane in three dimensional space is tangent to some surface at some point of that for |x−a| < R, then f is equal to the sum if its Taylor surface if they coincide at that point but have no series on |x − a| < R. other common points in some neighborhood of that A criteria showing when the remainder of the Taylor point. If the surface is given by a differentiable func- series converges to zero is given by the following Taytion z = f (x, y), then the tangent plane at the point lor’s inequality: (x0 , y0 , z0 ) is given by the equation If |f (N +1) (x)| ≤ M for |x−a| ≤ r, then the remainder of the Taylor series satisfies z − z0 = fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ). M tangent vector A vector (usually unit vector) in |RN (x) ≤ |x − a|N +1 , |x − a| ≤ d. (N + 1)! the direction of the tangent to a curve at some point. Tangent vector is orthogonal to the normal vector at the same point. If a space curve is given by the parametric vector function r(t), then its derivative r0 (t0 ),

The special case of Taylor series when a = 0 is called MacLaurin series. See corresponding entry for examples of Taylor-MacLaurin series.

144 Taylor’s inequality

See Taylor series.

term of a sequence In a sequence a1 , a2 , a3 , · · · each member (element, P∞entry) is called term. Similarly, in the series n=1 an each an is a term of the series. The same term(word) is used to describe the elements of any polynomial, which also are called monomials. terminal side Of two sides forming an angle in standard position, one is called the initial side and the other is the terminal. See the entry angle for all the details. term-by-term differentiation and integration P If a power series cn (x − a)n has radius of conver- time plot A type of graph, mostly used in Statistics, where the points are found from observations gence R > 0 then the function taken during some time period and then connected ∞ X by segments of straight line. n f (x) = c (x − a) n

n=0

is differentiable on the interval (a − R, a + R) and f 0 (x) =

∞ X

torus A geometric three dimensional surface that is the result of rotation of a circle around an axis not crossing or touching that circle.

ncn (x − a)n−1 ,

n=1

Z f (x)dx = C +

∞ X n=0

cn

(x − a)n+1 . n+1

The differentiated and integrated series have the same radius of convergence as the original series. tests of convergence of series Different tests to establish convergence or divergence of numeric or power series. Each of them have separate entries. See alternating series convergence test, comparison total differential For a function z = f (x, y) of two test, integral test, limit comparison test, ratio test, variables, the expression root test. ∂z ∂z dz = dx + dy. test of significance Statistical test, establishing ∂x ∂y the significance of the sample results. As a rule, this means that some test statistic should be compared Total differential for functions of three or more varito some pre-determined small value (called α-value). ables is defined in the similar way. The finding considered significant if the test statistic trace of a matrix For a square matrix A the trace (usually P-value) is smaller than this α-value. is the sum of all elements on the main diagonal. For tetrahedron A geometric object composed of four the matrix triangular faces, three of which meet at each vertex. 2 −2 0 5 −1 −4 A regular tetrahedron is one in which the four triangles are regular, or equilateral. 3 0 4

145 the trace is 2 + (−1) + 4 = 5. transcendental function Any function that is not algebraic. Transcendental functions include, but are not limited to, exponential, logarithmic, trigonometric, hyperbolic, and inverse trigonometric and hyperbolic functions. Besides, most of the functions defined with the use of power series or integrals are also transcendental. transcendental number A number that is not a solution of an algebraic equation with rational coefficients. Examples are the numbers e and π. transformation

transpose of a matrix The transpose of a matrix A is a matrix such that the rows are the columns of A and the columns are the rows of A. Hence, if A is a m × n matrix, then its transpose is a n × m matrix. The most common notation for the transpose is AT . Example: For the matrix 1 −2 0 −2 5 1 −4 3 3 0 1 2 the transpose is

See linear transformation.

transformations of a function The combined name for operations on functions including horizontal shift, vertical shift, reflection, stretching, shrinking (or compressing). The most general transformation of a given function f (x) would be the function g(x) = af (bx − c) + d. Here the constant b indicates the horizontal stretching (if |b| < 1) or shrinking (if |b| > 1). The number c/b indicates the horizontal shift, d is the vertical shift and the constant a is the vertical stretching (if |a| > 1) or shrinking (if |a| < 1). Also, multiplication by the constant a may result in reflection with respect to the x-axis if it is negative. Similarly, if b < 0 then the graph of the function f would be reflected with respect to the y-axis. A typical example of a function with all of the above transformations is the function

1 −2 0 −2

5 1 −4 3

3 0 . 1 2

transposition In any set of elements the operation that switches any two elements but leaves all the others unchanged is called a transposition. It is a special type of permutation. transverse axis of hyperbola The axis (line) passing through two foci of the hyperbola. trapezoid sides.

A quadrilateral with one pair of parallel

trapezoidal rule One of the methods of approximate integration. To calculate the approximate value Rb of the integral a f (x)dx using trapezoidal rule we y = −2 sin(3x + π/4) − 1. divide the interval [a, b] into n equal intervals of the length ∆x = (b − a)/n with the points x0 = a, x1 = The term translations of functions is also sometimes a + ∆x, x2 = a + 2∆x, · · · , xn = b. Then used. Z b ∆x transition matrix Suppose V is a vector space [f (x0 ) + 2f (x1 ) f (x)dx ≈ 0 0 0 0 2 and B = {u1 , u2 , · · · , un } and B = {u1 , u2 , · · · , un } a are two bases in that space. Let for some vector +2f (x2 ) + · · · + 2f (xn−1 ) + f (xn )]. v ∈ V [v]B and [v]B0 denote the coordinate matri0 ces of that vector with respect to the bases B and B For the degree of accuracy of this approximation see respectively. Then there exists a n × n matrix P such error estimate for the trapezoidal rule. that [v]B = P [v]B 0 . The matrix P is the transition triangle A geometric figure that consists of three matrix from the basis B to the basis B 0 and could be points (vertices) and three intervals connecting these written in the form points. Any two adjacent intervals form an angle, hence the name. Triangle is the simplest special case P = [[u01 ]B |[u02 ]B | · · · |[u0n ]B ] .

146 of a polygon. triangle inequality For any two vectors u and v in the Euclidean space Rn ||u + v|| ≤ ||u|| + ||v||,

segment will form some angle θ with the real axis, called the argument of z. The argument of a number can be found from the formula tan θ = b/a.The trigonometric form of the number z is z = r(cos θ + sin θ).

where the symbol || · || indicates the norm (length, magnitude) of a vector. The name comes from the fact that this inequality generalizes the geometric fact that any side of a triangle is shorter than the sum of the other two sides. The triangle inequality has further generalizations to the so-called metric spaces.

The Euler modification of this form is z = reiθ . Trigonometric form is convenient when multiplying, dividing, raising to whole power and especially for finding the roots of complex numbers. See multiplication of complex numbers, division of complex numbers, DeMoivre’s theorem,roots of a complex triangular matrix A matrix where all the ele- number. ments above or below the main diagonal are zeros. trigonometric functions The six functions See also lower triangular, upper triangular matrices. sin x, cos x, tan x, cot x, sec x and csc x. For trigonometric equations Equations involving trigonometric functions. Unlike identities, equations are not true statements for all or ”almost all” values of the variable, but might be true only for limited number of values if considered in finite interval. In cases when we are looking for solutions without restrictions on the interval, we usually get infinitely many solutions but they are always periodic repetitions of solutions on the finite interval. Examples: The equation sin x = 1/2

the definitions and graphs of each of them see the corresponding definitions.

has only two solutions x = π/6, x = 5π/6 on the interval [0, 2π). If considered on the real axis (−∞, ∞), if has infinitely many solutions found by adding arbitrary multiple of 2π to the two basic solutions: x = π/6 + 2πk, x = 5π/6 + 2πk, k = 0, ±1, ±2, · · ·. Similarly, the equation

cos(x − y) = cos x cos y + sin x sin y.

trigonometric identities Identities involving trigonometric functions. Identities can be true for all values of the variable(s) or for all values except where the functions are not defined. Trigonometric formulas are also identities. Examples: sin2 θ + cos2 θ = 1 cos x + sin x tan x = sec x

trigonometric integrals A group of indefinite integrals that contain different combinations of trigonometric functions. For most of these combinations there are methods to evaluate that integrals but not all trigonometric integrals can be expressed in the 2 4 cos x − 1 = 0 form of elementary functions. R n m has the solutions x = π/3, x = 2π/3 on [0, 2π) and 1) Integrals of the form sin x cos xdx. we get the general solution by adding a multiple of (a) If m = 2k + 1 is odd, then write Z 2π to that solutions. sinn x(cos2 x)k cos xdx trigonometric form of the complex number Let z = a + ib be a complex number. Then it can Z be represented on the complex plane as a point with = sinn x(1 − sin2 x)k cos xdx coordinates a and b. The distance of this point √ from the origin is the modulus of the number r = a2 + b2 . and using the substitution u = sin x reduce the inteR Now, if we connect the origin with the point z, that gral to algebraic: un (1 − u2 )k dx.

147 (b) If n = 2s + 1 is odd, then similar to the previous case Z (sin2 )s cosm x sin xdx Z = (1 − cos2 x)s cosm x sin xdx

Z = Z =−

(1 − u2 )2 u2 du = −

=− and the substitution u = cos x again reduces integral to algebraic. (c) If both n and m are even, then the use of halfangle formulas sin2 x =

1 1 (1 − cos 2x), cos2 x = (1 + cos 2x) 2 2

(possibly multiple times) reduces the integral to easy to evaluate form. R 2) Integrals of the form tann x secm xdx. (a) If m = 2k is even, then write Z tann x(sec2 )k−1 sec2 xdx Z =

Z

(sec2 x − 1)s secm−1 sec x tan xdx

and substitute u = sec x using the fact that (sec x)0 = sec x tan x. Once again, the integral reduces to algebraic. 3) In R evaluating theR integrals of the form sin nx cos mxdx, sin nx sin mxdx or R cos nx cos mxdx it is useful to use the product-tosum formulas. Many other trigonometric integrals can be calculated by different methods. Among trigonometric integrals that cannot be expressedR with the use of elementary functions is the integral sin(x2 )dx. Examples: Z Z 5 2 sin x cos xdx = (sin2 x)2 cos2 x sin xdx

u5 u7 u3 +2 − +C 3 5 7

cos3 x 2 cos5 x cos7 x + − + C. 3 5 7

Z Z

Z =

tan6 x sec4 xdx

tan6 x(1 + tan2 x) sec2 xdx

=

u6 (1 + u2 )du = =

u7 u9 + +C 7 9

tan7 x tan9 + + C. 7 9

Z

tann x(1 + tan2 x)k−1 sec2 xdx

and substitute u = tan x using the fact that (tan x)0 = sec2 x. Integral reduces to algebraic. (b) If n = 2s + 1 is odd, then write Z (tan2 x)s secm−1 x sec x tan xdx =

(1 − cos2 x)2 cos2 x sin xdx

sin 7x cos 4xdx 1 = 2

Z

Z sin 3xdx +

cos 11xdx

1 1 sin 11x + C. == − cos 3x + 6 22 trigonometric series A finite or infinite series of the form c+

∞ X

(an cos nθ + bn sin nθ).

n=1

In the case the coefficients c, an , bn have special relations to some function f (x), the series becomes a Fourier series. See the corresponding entry. trigonometric substitutions See integration by trigonometric substitution. trinomial A polynomial that has three terms. Trinomial might have one or more variables. Examples: 2x2 − 5x + 6, 3x3 y 2 + 4x2 y 4 − 6xy. triple product

Or scalar triple product of three

148

U

vectors u, v, w in three dimensional space is defined as u · (v × w) and could be calculated by the formula u1 u2 u3 u · (v × w) = v1 v2 v3 . unbiased statistic A sample statistic whose w1 w2 w3 sampling distribution has a mean value equal to the mean value of the population parameter that it is trivial solution An obvious, very simple solution estimating. Sample mean, proportion and variance of an equation or system of equations. As a rule, are unbiased statistics but the sample standard trivial solution is a zero solution. Any homogeneous deviation is not. linear system has a trivial solution. Many differen- unbounded function A function that is not tial equations have trivial (identically zero) solution bounded from above, below, or both. The functions along with non-zero solutions. f (x) = ln x, g(x) = x2 are unbounded but the function h(x) = sin x is bounded. twisted cubic A space curve given by the vector- unbounded region In case of a plane region, it function r(t) = (t, t2 , t3 ). is called bounded if it could be enclosed in some circle of finite radius centered at the origin. If there is no such circle, then the region is unbounded. Any region given by an inequality y > ax + b is an example of an unbounded region. Unbounded regions in the three dimensional space are defined similarly. undefined function A function that is not or cannot be defined for certain values of the variable. For example, the function f (x) = 1/x is undefined for x = 0 because division by zero is not defined. undercoverage (of population) In statistics, when sampling, sometimes certain parts of the population are covered less that the others. For example, when conducting telephone polls, people without phones are not covered at all. underdetermined linear system A system of linear equations that has more unknowns than equations. If this system is consistent, then it has infinitely many solutions. See also overdetermined linear systems. undetermined coefficients Also called the method of undetermined coefficients for certain types of equations. In using this method we either know or guess the form of the solution but do not know the value of the constant(s) and are trying to determine that value. One of the versions of

149 this method is described in the definition of partial fraction decomposition. Second application is for solution of non-homogeneous linear differential equations, working mainly for equations with constant coefficients. Suppose we are trying to find a particular solution of the equation

unit step function Similar to the Heaviside function, defined by the formula n 1 if t ≥ c uc (t) = 0 if t < c for c ≥ 0. The Laplace transform of this function is the function L[uc (t)] = e−cs /s, s > 0.

unit circle A circle with radius equal to one. In Cartesian coordinate system such a circle could be given by the equation (x − a)2 + (y − b)2 = 1, where We guess that the solution should have the form (a, b) is the coordinate of the center. The picture Y (t) = Ae2t , where A is yet to be determined shows the unit circle with center at the origin. coefficient. After we plug-in this supposed solution into the equation we get (4A − 6A − 4A)e2t = 3e2t and A = −1/2 and the coefficient is found. Similar method also works for the systems of linear equations where, of course, we have to solve systems of linear algebraic equations with respect to a number of undetermined coefficients. y 00 − 3y 0 − 4y = 3e2t .

unit sphere A sphere with radius one. In Cartesian coordinate system the equation is x2 + y 2 + z 2 = uniform distribution A continuous probability 1. The notion of the sphere extends also to higher distribution where the density curve is a function that dimensional Euclidean spaces. is constant on some interval [a, b] and is zero everywhere else. Because of the requirement that the area unit vector A vector that has norm (length, magunder the curve should be 1, the height of the line is nitude) equal to 1. If v is an arbitrary non-zero vector in the vector space V , then the vector u = v/||v|| has 1/(b − a). unit norm and hence, is a unit vector. See also norunion of sets If A and B are two sets of some mal vector, tangent vector. objects, then their union A ∪ B is defined to be the set of all elements that belong to A or B or unitary diagonalization Suppose A is some both of them. Example: If A = {1, 3, 5, 7} and square matrix and P is a unitary matrix of the same B = {2, 4, 6, 8} then A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}. size. Then A−1is said to be unitary diagonalizable if In case when the two sets have common elements, the matrix P AP is diagonal. then in the union these elements are counted only unitary matrix A matrix A with complex entries once. Hence, if A = {1, 2, 3} and B = {2, 3, 4}, then is unitary, if its inverse A−1 is equal to its conjugate A ∪ B = {1, 2, 3, 4}. transpose A∗ . The matrix uniqueness theorems See existence and unique1 1+i 1+i . ness theorems. 2 1−i 1−i

150 is a unitary matrix.

V

unlike fractions Two (or more) fractions that have different denominators. The fractions 43 and 47 are unlike fractions. On the other hand, the fractions 3 6 value of a function The numeric expression of a 8 and 16 are not unlike fractions because the second (1) The one could be reduced to simplest terms as 38 with the given function at a given point. 3Examples: 2 value of the function f (x) = 3x − 2x + 5x − 1 at same denominator. the point x = 2 is determined by substituting the unlike terms Two terms in an algebraic expression value x = 2 in the expression of the function: f (2) = that do not have identical variable parts (factors). In 3 · 23 − 2 · 22 + 5 · 2 − 1 = 25. (2) The value of the the expression 2xy + 2xy 2 the two terms are unlike function f (x) = sin x at the point x = π/6 is 1/2. terms, because xy and xy 2 are not identical. The To calculate the values of more complicated functions constant factors do not play role in deciding if the approximate methods and/or calculators are needed. terms are like terms or unlike terms. variable One of the basic mathematical objects upper triangular matrix A square matrix where along with constants, functions, etc. Variables are all the entries below the main diagonal are zeros. Ex- quantities that do not have fixed value (as the ample: constants) but rather have the ability to change. For 2 4 2 example, time is a variable quantity because it does 0 1 −3 . not stand still. Variables are denoted by symbols 0 0 3 such as x, y, z, t, u, and others. In certain situations The determinant of such a matrix is just the product there might be two or more variable quantities of all diagonal elements. See also lower diagonal and if they are related to each other by some rule or formula, we call that a function or relation. matrix. Accordingly, the variables in that relations would be independent or dependent. See also the entries explanatory, quantitative, qualitative, response, random, lurking variables. variance For a set of numeric values x1 , x2 , x3 , · · · , xn with the mean µ, the variance is defined by the formula Pn (xi − µ)2 2 σ = i=1 . n−1 The square root of this quantity is called standard deviation. variation Specific types of relationships between two or more variable. (1) Direct variation. In this case one of the variables is directly proportional to another variable in some form. For example if y = kx (k is some numeric constant) we say that y varies directly with x. If y = kx2 then y varies directly with x2 . (2) Inverse variation. In this case one variable is inversely proportional to some form of the other vari-

151 able. If y =√k/x we say that y varies inversely √ with x. If y = k/ x, then y varies inversely with x, and so on. (3) Joint variation. In this case one variable is directly or inversely proportional to some forms of two or more other variable. Example: z = kx2 /y 3 .

we notice that the corresponding homogeneous equation y 00 + 4y = 0 has the general solution y(t) = c1 cos 2t + c2 sin 2t and look for a particular solution of the non-homogeneous equation in the form Y (t) = u1 (t) cos 2t + u2 (t) sin 2t. Now, the Wronskian of this equation is W = 2 and by the Theorem above, u1 (t) = −3 cos t and u2 (t) = 3/2 ln | csc t − variation of parameters A method of finding a cot t| + 3 cos t. Plugging in these functions into the particular solution for non-homogeneous linear differprevious formula we find a particular solution of the ential equations. This method is the development of non-homogeneous equation: the method of undetermined coefficients and allows to find particular solutions for much wider classes of Y (t) = −3 sin t cos 2t + 3 cos t sin 2t equations. Suppose we want to find the general solution of the 3 + sin 2t ln | csc t − cot t|. equation 2 00 0 Similar method works also for equations of higher ory + p(t)y + q(t)y = g(t). (1) der and for systems of non-homogeneous equations. First of all, the general solution of this equation is vector One of the basic mathematical objects, vecthe combination of the general solution of the corretors could be viewed as quantities that have both sponding homogeneous equation some numeric value and a direction. Many physical notions (such as force or pressure) have that propery 00 + p(t)y 0 + q(t)y = 0 (2) ties and vectors are abstract mathematical represenand any particular solution of the equation (1). As- tations of these facts. As with the other mathematisume also that we know the general solution of the cal objects (numbers, variables, functions, matrices), homogeneous equation (2) and it is many operations are possible to perform with vectors too. As objects, vectors may exist on the plane, three yh (t) = c1 y1 (t) + c2 y2 (t), dimensional space, or more generally, in any vector where y1 , y2 are two linearly independent solutions. space. The idea behind the method of variation of parame- 1) Addition of vectors in the plane is defined by the ters is to replace constants c1 , c2 by some, yet to be parallelogram rule (see addition and subtraction of determined functions u1 (t), u2 (t) and try to find a vectors). More generally, if v and u are two vectors particular solution in the form in some vector space V , then Y (t) = u1 (t)y1 (t) + u2 (t)y2 (t).

v + u = (v1 + u1 , v2 + u2 , · · · , vn + un ),

Theorem. The particular solution of the equation (1) could be written in the form Z t y2 (s)g(s) ds Y (t) = −y1 (t) W (y1 , y2 )(s) t0 Z t y1 (s)g(s) +y2 (t) ds, t0 W (y1 , y2 )(s)

where v = (v1 , v2 , · · · , vn ), u = (u1 , u2 , · · · , un ). 2) The scalar product of a number c and the vector v is defined as cv = (cv1 , cv2 , · · · , cvn ). 3) It is impossible to define a product of two vectors in any space V in such a manner that the result be another vector from the same space and at the same time satisfy all the expected properties of multiplication (commutative, associative, distributive). Instead, the inner product (or dot product) of two vectors could be defined which is not a vector but a number (real or complex, depending on the space

where W (y1 , y2 ) is the Wronskian of y1 and y2 . Example: To solve the equation y 00 + 4y = 3 csc t

152 V ): v · u = v1 u1 + v2 u2 + · · · + vn un . See also cross product for one specific type of vector multiplication. 4) The length (or magnitude, or norm) of a vector is √ now defined as ||v|| = v · v. 5) The angle between two vectors now could be defined with the help of inner product as follows:

to V ; (2) u + v = v + u; (3) u + (v + w) = (u + v) + w; (4) There is a vector 0 in V , named zero vector, such that u + 0 = 0 + u = u; (5) For each u in V there is a vector −u also in V , such that u + (−u) = (−u) + u = 0; v·u (6) If u is in V then ku is also in V ; cos θ = . ||v||||u|| (7) k(u + v) = ku + kv; Two vectors are called parallel if the angle between (8) (k + `)u = ku + `u; them is either zero or π(= 180◦ ). In case when the (9) k(`u) = (k`)u; angle is π/2 = 90◦ , the vectors are called perpendic- (10) 1u = u. If the scalars k, `, ... are real then the vector space is ular, or orthogonal. For additional facts about vectors see also basis, lin- called real vector space. In case when we chose comear dependence and independence of vectors, tangent plex scalars, the space is called complex vector space. Among the most common examples of vector spaces vector, unit vector, zero vector. are the Euclidean space Rn , the space of all m × n vector field Let D be a region on the plane. A vec- matrices, the space of all polynomials of degree n or tor field F is a function that maps any point (x, y) less, and many others. from D to another point on the plane. This vector Venn diagram A graphical method of representfield could be written now as ing sets and their relations. Used in set theory, logic, statistics, and probability theory. Especially helpful F(x, y) = P (x, y)i + Q(x, y)j, in understanding the union, intersection of sets. Bewhere P, Q are ordinary functions of two variables. low is a typical Venn diagram of intersection of two Vector fields can be defined also for three or any sets. higher number of variables in exactly the same way. See also curl of the vector field, flux, gradient vector field. vector function A vector such that the coordinates (components) are functions. Example: F(t) = (f (t), g(t), q(t)) = (sin t, cos t, t). vector product The product of two vectors is impossible to define in a way that the result be another vector from the same space with expected properties of multiplication as we have them for numbers, vertex A term used in many different situations variables, or functions. For special forms of vector with slightly different meaning. In some sense verproducts see cross product, inner product. tices are the ”extreme points” of geometric objects vector space The abstract vector space V is a col- and figures. For example, the vertex of an angle is the lection of objects called vectors for which the op- point from where the two rays forming the angle come erations of vector addition and scalar multiplication out (see also angle). For a polygon or a polytope the are defined. Denoting the vectors by u, v, w, ... and vertices are the ”corner” points of that figures. At scalars by k, `, ... we require that they satisfy the fol- the same time the notion of the vertex also applies to other geometric figures, such as ellipses, parabolowing list of axioms of the vector space: (1) If u and v belong to V , then u + v also belongs las, and hyperbolas. See the corresponding entries

153 for exact descriptions.

of volumes of more complicated solids it is necessary to involve calculus. In general, if a solid is defined vertical asymptote The (vertical) line x = a is a by a function z = f (x, y), where the variables (x, y) vertical asymptote for a function f (x) if that funcare changing on some rectangle D = [a, b] × [c, d], tion grows unboundedly as the point x approaches then the volume of that solid is given by the double the point a. In terms of limits this property is preintegral Z bZ d sented by one of the following limit conditions: f (x, y)dxdy. V = lim f (x) = ∞, lim f (x) = ∞ c a x→a x→a− In cases when the solid is a solid of revolution of some lim f (x) = ∞, lim f (x) = −∞ curve about some axis, or if the solid has some other x→a x→a+ extra properties of symmetry, the volume could be lim− f (x) = −∞, lim+ f (x) = −∞. calculated by the method of cylindrical shells or discs x→a x→a (washers). These conditions mean that the function cannot have a finite (specific) value at that point. As a result, the function’s graph can never cross the vertical asymptote. Examples: (1) For the function f (x) = 1/x2 the line x = 0 (the y-axis) is the only vertical asymptote. (2) The function f (x) = x2 /(x − 1)(x + 2) has vertical asymptotes x = 1 and x = −2. vertical line test A geometric method of determining if a given graph is a graph of a function or a relation. If any vertical line crosses the graph only once, then it is a function, because that means that to each value of x there is only one corresponding value of y. If this condition is violated at even one point we have a relation instead of a function. This method is not precise because it depends on graphing skills that are subjective, still it is a useful visual tool. vertical shifts of graphs Addition or subtraction of any constant to any given function results in the vertical shift (up or down) of the graph of that function. Example: f (x) = sin x,

g(x) = sin x + 2.

The graphs of f and g are identical except that the second one takes values two more than the first one. Its graph is two units above the graph of f . volume A geometric notion indicating the amount of space occupied by an object in three-dimensional space. The volumes of simple geometric solids, such as prisms, pyramids, spheres, cylinders, cones, and others are known for a long time. For calculations

154

W

Wronskian Consider the second order linear homogeneous equation y 00 + p(x)y 0 + q(x)y = 0

wave equation For differential equations. In case with the initial value conditions y(t0 ) = y0 , y 0 (t0 ) = of a function of two spacial variables x, y, and the y00 . Assume also that y1 and y2 are two solutions of time variable t, the equation the equation (without initial conditions). The determinant 2 2 2 ∂ u(x, y, t) ∂ u(x, y, t) ∂ u(x, y, t) y1 (t0 ) y2 (t0 ) = + . det ∂t2 ∂x2 ∂y 2 y10 (t0 ) y20 (t0 ) is called the Wronskian of the equation and plays important role in finding the general solution of the equation. See Abel’s formula for the presentation of weighted mean Or weighted average. In cases the Wronskian and variation of parameters for an apwhen different numbers have different weights, their plication. Wronskian also generalizes to the case of average or mean is calculated using the weighted homogeneous equations of arbitrary order n. For the mean formula. Let w1 , w2 , · · · , wn be all positive equation numbers (some may be zero, but not all) and assume that x1 , x2 , · · · , xn are numeric values with cory (n) + p1 (x)y (n−1) + · · · + pn−1 (x)y 0 + pn (x)y = 0 responding weights. This means that the value xi has the weight wi , 1 ≤ i ≤ n. Then the weighted mean with solutions y1 , y2 , · · · , yn , the Wronskian is the deis terminant w1 x1 + w2 x2 + · · · + wn xn xw = . y y2 . . . yn 1 w1 + w2 + · · · + wn 0 0 0 y2 . . . yn y1 In cases when the det .. .. Pn sum of wi is 1, the formula sim .. plifies to x = w x . Example: Assume a stu. . . In the simpler case of one spacial variable x and time variable t the equation has the form a2 uxx = utt .

w

i=1

i i

dent get the scores 62, 78, 81, 92 on her tests, 84 on the Final, and 90 for her homework. Each test is valued at 15%, the Final is 30% and homework is valued at 10%. Then the mean score of the student will be 0.15(62 + 78 + 81 + 92) + 0.3 · 84 + 0.1 · 90 = 72.95, because the sum of all weights is one (0.15+0.15+0.15+0.15+0.3+0.1=1). weighted inner product Let u = (u1 , u2 , · · · , un ) and v = (v1 , v2 , · · · , vn ) be two vectors in the Euclidean space Rn and assume w1 , w2 , · · · , wn is a sequence of positive numbers. Then the weighted inner product with this sequence (weight) is hu, viw = w1 u1 v1 + w2 u2 v2 + · · · + wn un vn . whole number The set of the whole numbers consists of all natural numbers with addition of the number zero. Hence, the whole numbers are: 0, 1, 2, 3, 4, · · ·

(n−1)

y1

(n−1)

y2

...

(n−1)

yn

155

XYZ x-axis In Cartesian coordinate system on the plane the horizontal axis. The projection of any point on the plane onto the x-axis gives the x-coordinate of the point, always written first. For example, for the point (5, −3) the first entry 5 is the x-coordinate.

for the value x = 0 then it always has the y-intercept. Unlike x-intercepts, a function may have no more than one y-intercept. On the other hand, a relation may have more than one y-intercept. Examples: (1) The function f (x) = x3 + 3x2 − 4x − 12 has the value f (0) = −12 and that is the only y-intercept of that function; (2) The function f (x) = 1/x is not defined for x = 0 and, as a result, does not have a y-intercept; (3) The relation x2 + y 2 = 1 representing the unit circle has two y-intercepts at the points (0, 1) and (0, −1). z-axis In three-dimensional Cartesian coordinate system, in addition to x- and y-axes, there is also the z-axis, that is perpendicular to the plane formed by these two axes. Any point in the space has three orthogonal projections, corresponding to three axes. The projection onto the z-axis gives the z-coordinate of the point that is always written on the third position: For the point (1,2,3) the third entry 3 is the z-coordinate.

x-coordinate

See above, x-axis.

x-intercept The intersection of the graph of any function with the x-axis. For polynomials, trigonometric, logarithmic, exponential and other functions, these points coincide with the zeros of the corresponding types of equations. Examples: (1) To find the x-intercepts of the function f (x) = x3 + 3x2 − 4x − 12 we solve the corresponding algebraic equation x3 + 3x2 − 4x − 12 = 0. This equation has the zeros x = 2, −2, −3 and that points are exactly the x-intercepts of the function f (x); (2) The function f (x) = sin 2x has the intercepts x = πn/2, n = 0, ±1, ±2, · · · because they are the solutions of the corresponding trigonometric equation sin 2x = 0. y-axis In Cartesian coordinate system on the plane the vertical axis. The projection of any point on the plane onto the y-axis gives the y-coordinate of the point, always written second. For example, for the point (5, −3) the second entry -3 is the y-coordinate. See the figure above. y-coordinate

See above, y − axis.

y-intercept The intersection of the graph of any function with the y-axis. If that function is defined

z-coordinate See above, z-axis. z-score Suppose x is a random variable and assume x ¯ is its mean and s its standard deviation. The quantity z = (x − x ¯)/s is the z-score (or standardized value) of the variable x. If the variable x belongs to a normal distribution with the mean x ¯ and standard deviation s, then the variable z (the z-score) belongs to standard normal distribution: its mean is zero and standard deviation is 1. The z-score measures the distance of the value x from the mean in terms of standard deviation units. z-scores are also sometimes called z statistic. zero Originally, one of the integers in the real number system that is smaller that 1 but greater than −1. The only real number that is neither positive nor negative. The most important property of zero is that it is the additive identity which is expressed algebraically as a + 0 = a. The importance of the number zero goes beyond the real numbers because it serves as the additive identity also for the complex numbers: If z is any complex number, then z + 0 = z. Also, zero could be considered as the ”zero function” that for all values of the variable x equals to zero. In this interpretation zero

156 is also the additive identity in the set of all functions. See also zero matrix. zero factor property Also called zero product property. If for two numbers (real or complex) a and b, a · b = 0, then either a = 0 or b = 0 or both are zero. This property of numbers is one of the most important tools in solving algebraic or trigonometric equations. Examples: (1) x3 +3x2 −4x−12 = 0. Factoring the polynomial on the left side by grouping, we get the equation (x − 2)(x + 2)(x + 3) = 0 which by the zero factor property results in three linear equations x − 2 = 0, x + 2 = 0, x + 3 = 0 with the solutions x = 2, x = −2, x = −3. (2) sin 2x − cos x = 0. Using the double angle formula for the sine function and factoring we get cos x(2 sin x − 1) = 0 and again, by the zero factor property this results in two simpler equations cos x = 0 and sin x = 1/2. The solutions on the base interval [0, 2π) are: x = π/2, 3π/2, π/6, 5π/6. The zero factor property is not true for other mathematical object, for example for matrices: If the product A · B of two matrices is the zero matrix then none of the matrices A or B are necessarily the zero matrices themselves. zero matrix A matrix of arbitrary size m × n such that all of the entries are zeros. Example: 0 0 0 0 0 0 0 0. 0 0 0 0 This matrix is the additive identity in the set of all matrices of the same size: Adding this matrix to any other matrix does not change it. zeros of the equation Also called roots or solutions of the equation. Any real (or complex) number that substituted into the equation results in numeric identity. The number x = 2 is the zero of the equation x2 − x − 2 = 0 because substituting that value into the left side of the equation results in numeric identity 0 = 0. Every polynomial equation of degree n ≥ 1 has exactly n zeros, if we accept complex zeros and count them according to multiplicity. See also Fundamental Theorem of Algebra. zero subspace The subspace of a vector space that consists of a single zero vector x = 0. Zero subspace

is the part of any vector space. zero transformation The (linear) transformation between any vector spaces V and W such that T (u) = 0 for any vector u ∈ V . zero vector The vector that has all zero components: 0 = (0, 0, · · · , 0). This vector plays the role of the additive unity for any vector space: x + 0 = 0 + x = x for any vector x of the vector space.

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