Most of the terms listed in Wikipedia glossaries are already defined and explained within Wikipedia itself. However, glossaries like this one are useful for looking up, comparing and reviewing large numbers of terms together. You can help enhance this page by adding new terms or writing definitions for existing ones. This glossary of calculus is a list of definitions about calculus, its sub-disciplines, and related fields.
A
Abel's test A method of testing for the convergence of an infinite series. absolute convergence An infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0 ∞ a n {\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}} is said to converge absolutely if ∑ n = 0 ∞ | a n | = L {\displaystyle \textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=L} for some real number L {\displaystyle \textstyle L} . Similarly, an improper integral of a function, ∫ 0 ∞ f ( x ) d x {\displaystyle \textstyle \int _{0}^{\infty }f(x)\,dx} , is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if ∫ 0 ∞ | f ( x ) | d x = L . {\displaystyle \textstyle \int _{0}^{\infty }\left|f(x)\right|dx=L.}
absolute maximum The highest value a function attains. absolute minimum The lowest value a function attains. absolute value The absolute value or modulus |x| of a real number x is the non-negative value of x without regard to its sign. Namely, |x| = x for a positive x, |x| = −x for a negative x (in which case −x is positive), and |0| = 0. For example, the absolute value of 3 is 3, and the absolute value of −3 is also 3. The absolute value of a number may be thought of as its distance from zero. alternating series An infinite series whose terms alternate between positive and negative. alternating series test Is the method used to prove that an alternating series with terms that decrease in absolute value is a convergent series. The test was used by Gottfried Leibniz and is sometimes known as Leibniz's test, Leibniz's rule, or the Leibniz criterion. annulus A ring-shaped object, a region bounded by two concentric circles. antiderivative An antiderivative, primitive function, primitive integral or indefinite integral of a function f is a differentiable function F whose derivative is equal to the original function f. This can be stated symbolically as F ′ = f {\displaystyle F'=f} . The process of solving for antiderivatives is called antidifferentiation (or indefinite integration) and its opposite operation is called differentiation, which is the process of finding a derivative. arcsin
area under a curve
asymptote In analytic geometry, an asymptote of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. Some sources include the requirement that the curve may not cross the line infinitely often, but this is unusual for modern authors. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity. automatic differentiation In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation or computational differentiation, is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.
average rate of change
B
binomial coefficient Any of the positive integers that occurs as a coefficient in the binomial theorem is a binomial coefficient. Commonly, a binomial coefficient is indexed by a pair of integers n ≥ k ≥ 0 and is written ( n k ) . {\displaystyle {\tbinom {n}{k}}.} It is the coefficient of the xk term in the polynomial expansion of the binomial power (1 + x)n, and it is given by the formula
( n k ) = n ! k ! ( n − k ) ! . {\displaystyle {\binom {n}{k}}={\frac {n!}{k!(n-k)!}}.}
binomial theorem (or binomial expansion) Describes the algebraic expansion of powers of a binomial. bounded function A function f defined on some set X with real or complex values is called bounded, if the set of its values is bounded. In other words, there exists a real number M such that
| f ( x ) | ≤ M {\displaystyle |f(x)|\leq M}
for all x in X. A function that is not bounded is said to be unbounded.
Sometimes, if f(x) ≤ A for all x in X, then the function is said to be bounded above by A. On the other hand, if f(x) ≥ B for all x in X, then the function is said to be bounded below by B. bounded sequence .
C
calculus (From Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus) is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. Cavalieri's principle Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes. chain rule The chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f and g are functions, then the chain rule expresses the derivative of their composition f ∘ g (the function which maps x to f(g(x)) ) in terms of the derivatives of f and g and the product of functions as follows:
( f ∘ g ) ′ = ( f ′ ∘ g ) ⋅ g ′ . {\displaystyle (f\circ g)'=(f'\circ g)\cdot g'.}
This may equivalently be expressed in terms of the variable. Let F = f ∘ g, or equivalently, F(x) = f(g(x)) for all x. Then one can also write
F ′ ( x ) = f ′ ( g ( x ) ) g ′ ( x ) . {\displaystyle F'(x)=f'(g(x))g'(x).}
The chain rule may be written in Leibniz's notation in the following way. If a variable z depends on the variable y, which itself depends on the variable x, so that y and z are therefore dependent variables, then z, via the intermediate variable of y, depends on x as well. The chain rule then states,
d z d x = d z d y ⋅ d y d x . {\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}.}
The two versions of the chain rule are related; if z = f ( y ) {\displaystyle z=f(y)} and y = g ( x ) {\displaystyle y=g(x)} , then
d z d x = d z d y ⋅ d y d x = f ′ ( y ) g ′ ( x ) = f ′ ( g ( x ) ) g ′ ( x ) . {\displaystyle {\frac {dz}{dx}}={\frac {dz}{dy}}\cdot {\frac {dy}{dx}}=f'(y)g'(x)=f'(g(x))g'(x).}
In integration, the counterpart to the chain rule is the substitution rule. change of variables Is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem. cofunction A function f is cofunction of a function g if f(A) = g(B) whenever A and B are complementary angles. This definition typically applies to trigonometric functions. The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620). concave function Is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap or upper convex. constant of integration The indefinite integral of a given function (i.e., the set of all antiderivatives of the function) on a connected domain is only defined up to an additive constant, the constant of integration. This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function f ( x ) {\displaystyle f(x)} is defined on an interval and F ( x ) {\displaystyle F(x)} is an antiderivative of f ( x ) {\displaystyle f(x)} , then the set of all antiderivatives of f ( x ) {\displaystyle f(x)} is given by the functions F ( x ) + C {\displaystyle F(x)+C} , where C is an arbitrary constant (meaning that any value for C makes F ( x ) + C {\displaystyle F(x)+C} a valid antiderivative). The constant of integration is sometimes omitted in lists of integrals for simplicity. continuous function Is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism. continuously differentiable A function f is said to be continuously differentiable if the derivative f′(x) exists and is itself a continuous function. contour integration In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. convergence tests Are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} . convergent series In mathematics, a series is the sum of the terms of an infinite sequence of numbers.
Given an infinite sequence ( a 1 , a 2 , a 3 , … ) {\displaystyle \left(a_{1},\ a_{2},\ a_{3},\dots \right)} , the nth partial sum S n {\displaystyle S_{n}} is the sum of the first n terms of the sequence, that is,
S n = ∑ k = 1 n a k . {\displaystyle S_{n}=\sum _{k=1}^{n}a_{k}.}
A series is convergent if the sequence of its partial sums { S 1 , S 2 , S 3 , … } {\displaystyle \left\{S_{1},\ S_{2},\ S_{3},\dots \right\}} tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases. More precisely, a series converges, if there exists a number ℓ {\displaystyle \ell } such that for any arbitrarily small positive number ε {\displaystyle \varepsilon } , there is a (sufficiently large) integer N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq \ N} ,
| S n − ℓ | ≤ ε . {\displaystyle \left|S_{n}-\ell \right\vert \leq \ \varepsilon .}
If the series is convergent, the number ℓ {\displaystyle \ell } (necessarily unique) is called the sum of the series.
Any series that is not convergent is said to be divergent. convex function In mathematics, a real-valued function defined on an n-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. For a twice differentiable function of a single variable, if the second derivative is always greater than or equal to zero for its entire domain then the function is convex. Well-known examples of convex functions include the quadratic function x 2 {\displaystyle x^{2}} and the exponential function e x {\displaystyle e^{x}} . Cramer's rule In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution. It expresses the solution in terms of the determinants of the (square) coefficient matrix and of matrices obtained from it by replacing one column by the column vector of right-hand-sides of the equations. It is named after Gabriel Cramer (1704–1752), who published the rule for an arbitrary number of unknowns in 1750, although Colin Maclaurin also published special cases of the rule in 1748 (and possibly knew of it as early as 1729). critical point A critical point or stationary point of a differentiable function of a real or complex variable is any value in its domain where its derivative is 0. curve A curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight. curve sketching In geometry, curve sketching (or curve tracing) includes techniques that can be used to produce a rough idea of overall shape of a plane curve given its equation without computing the large numbers of points required for a detailed plot. It is an application of the theory of curves to find their main features. Here input is an equation. In digital geometry it is a method of drawing a curve pixel by pixel. Here input is an array (digital image).
D
damped sine wave Is a sinusoidal function whose amplitude approaches zero as time increases. degree of a polynomial Is the highest degree of its monomials (individual terms) with non-zero coefficients. The degree of a term is the sum of the exponents of the variables that appear in it, and thus is a non-negative integer. derivative The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances. derivative test A derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function. differentiable function A differentiable function of one real variable is a function whose derivative exists at each point in its domain. As a result, the graph of a differentiable function must have a (non-vertical) tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps. differential (infinitesimal) The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity. For example, if x is a variable, then a change in the value of x is often denoted Δx (pronounced delta x). The differential dx represents an infinitely small change in the variable x. The idea of an infinitely small or infinitely slow change is extremely useful intuitively, and there are a number of ways to make the notion mathematically precise.
Using calculus, it is possible to relate the infinitely small changes of various variables to each other mathematically using derivatives. If y is a function of x, then the differential dy of y is related to dx by the formula
d y = d y d x d x , {\displaystyle dy={\frac {dy}{dx}}\,dx,}
where dy/dx denotes the derivative of y with respect to x. This formula summarizes the intuitive idea that the derivative of y with respect to x is the limit of the ratio of differences Δy/Δx as Δx becomes infinitesimal. differential calculus Is a subfield of calculus concerned with the study of the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calculus, the study of the area beneath a curve. differential equation Is a mathematical equation that relates some function with its derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the equation defines a relationship between the two. differential operator . differential of a function In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by
d y = f ′ ( x ) d x , {\displaystyle dy=f'(x)\,dx,}
where f ′ ( x ) {\displaystyle f'(x)} is the derivative of f with respect to x, and dx is an additional real variable (so that dy is a function of x and dx). The notation is such that the equation
d y = d y d x d x {\displaystyle dy={\frac {dy}{dx}}\,dx}
holds, where the derivative is represented in the Leibniz notation dy/dx, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes
d f ( x ) = f ′ ( x ) d x . {\displaystyle df(x)=f'(x)\,dx.}
The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. Traditionally, the variables dx and dy are considered to be very small (infinitesimal), and this interpretation is made rigorous in non-standard analysis. differentiation rules . direct comparison test A convergence test in which an infinite series or an improper integral is compared to one with known convergence properties. Dirichlet's test Is a method of testing for the convergence of a series. It is named after its author Peter Gustav Lejeune Dirichlet, and was published posthumously in the Journal de Mathématiques Pures et Appliquées in 1862. The test states that if { a n } {\displaystyle \{a_{n}\}} is a sequence of real numbers and { b n } {\displaystyle \{b_{n}\}} a sequence of complex numbers satisfying
a n + 1 ≤ a n {\displaystyle a_{n+1}\leq a_{n}}
lim n → ∞ a n = 0 {\displaystyle \lim _{n\rightarrow \infty }a_{n}=0}
| ∑ n = 1 N b n | ≤ M {\displaystyle \left|\sum _{n=1}^{N}b_{n}\right|\leq M} for every positive integer N
where M is some constant, then the series
∑ n = 1 ∞ a n b n {\displaystyle \sum _{n=1}^{\infty }a_{n}b_{n}}
converges. disc integration Also known in integral calculus as the disc method, is a means of calculating the volume of a solid of revolution of a solid-state material when integrating along an axis "parallel" to the axis of revolution. divergent series Is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit. discontinuity Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. dot product In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. In Euclidean geometry, the dot product of the Cartesian coordinates of two vectors is widely used and often called "the" inner product (or rarely projection product) of Euclidean space even though it is not the only inner product that can be defined on Euclidean space; see also inner product space. double integral The multiple integral is a definite integral of a function of more than one real variable, for example, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in R2 are called double integrals, and integrals of a function of three variables over a region of R3 are called triple integrals.
E
e (mathematical constant) The number e is a mathematical constant that is the base of the natural logarithm: the unique number whose natural logarithm is equal to one. It is approximately equal to 2.71828, and is the limit of (1 + 1/n)n as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series
e = ∑ n = 0 ∞ 1 n ! = 1 1 + 1 1 + 1 1 ⋅ 2 + 1 1 ⋅ 2 ⋅ 3 + ⋯ {\displaystyle e=\displaystyle \sum \limits _{n=0}^{\infty }{\dfrac {1}{n!}}={\frac {1}{1}}+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots }
elliptic integral In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse. They were first studied by Giulio Fagnano and Leonhard Euler (c. 1750). Modern mathematics defines an "elliptic integral" as any function f which can be expressed in the form
f ( x ) = ∫ c x R ( t , P ( t ) ) d t , {\displaystyle f(x)=\int _{c}^{x}R\left(t,{\sqrt {P(t)}}\right)\,dt,}
where R is a rational function of its two arguments, P is a polynomial of degree 3 or 4 with no repeated roots, and c is a constant.. essential discontinuity For an essential discontinuity, only one of the two one-sided limits needs not exist or be infinite. Consider the function
f ( x ) = { sin 5 x − 1 for x < 1 0 for x = 1 1 x − 1 for x > 1 {\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\{\frac {1}{x-1}}&{\mbox{ for }}x>1\end{cases}}}
Then, the point x 0 = 1 {\displaystyle \scriptstyle x_{0}\;=\;1} is an essential discontinuity.
In this case, L − {\displaystyle \scriptstyle L^{-}} doesn't exist and L + {\displaystyle \scriptstyle L^{+}} is infinite – thus satisfying twice the conditions of essential discontinuity. So x0 is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from the term essential singularity which is often used when studying functions of complex variables. Euler method Euler's method is a numerical method to solve first order first degree differential equation with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who treated it in his book Institutionum calculi integralis (published 1768–1870). exponential function In mathematics, an exponential function is a function of the form
where b is a positive real number, and in which the argument x occurs as an exponent. For real numbers c and d, a function of the form f ( x ) = a b c x + d {\displaystyle f(x)=ab^{cx+d}} is also an exponential function, as it can be rewritten as
a b c x + d = ( a b d ) ( b c ) x . {\displaystyle ab^{cx+d}=\left(ab^{d}\right)\left(b^{c}\right)^{x}.}
extreme value theorem States that if a real-valued function f is continuous on the closed interval [a,b], then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in [a,b] such that:
f ( c ) ≥ f ( x ) ≥ f ( d ) for all x ∈ [ a , b ] . {\displaystyle f(c)\geq f(x)\geq f(d)\quad {\text{for all }}x\in [a,b].}
A related theorem is the boundedness theorem which states that a continuous function f in the closed interval [a,b] is bounded on that interval. That is, there exist real numbers m and M such that:
m < f ( x ) < M for all x ∈ [ a , b ] . {\displaystyle m<f(x)<M\quad {\text{for all }}x\in [a,b].}
The extreme value theorem enriches the boundedness theorem by saying that not only is the function bounded, but it also attains its least upper bound as its maximum and its greatest lower bound as its minimum. extremum In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given range (the local or relative extrema) or on the entire domain of a function (the global or absolute extrema). Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.
As defined in set theory, the maximum and minimum of a set are the greatest and least elements in the set, respectively. Unbounded infinite sets, such as the set of real numbers, have no minimum or maximum.
F
Faà di Bruno's formula Is an identity in mathematics generalizing the chain rule to higher derivatives, named after Francesco Faà di Bruno (1855, 1857), though he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French mathematician Louis François Antoine Arbogast stated the formula in a calculus textbook, considered the first published reference on the subject.
Perhaps the most well-known form of Faà di Bruno's formula says that
d n d x n f ( g ( x ) ) = ∑ n ! m 1 ! 1 ! m 1 m 2 ! 2 ! m 2 ⋯ m n ! n ! m n ⋅ f ( m 1 + ⋯ + m