Glossary of calculus
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Short description: List of definitions of terms and concepts commonly used in calculus
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This glossary of calculus is a list of definitions about calculus, its sub-disciplines, and related fields.
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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 [math]\displaystyle{ \textstyle\sum_{n=0}^\infty a_n }[/math] is said to converge absolutely if [math]\displaystyle{ \textstyle\sum_{n=0}^\infty \left|a_n\right| = L }[/math] for some real number [math]\displaystyle{ \textstyle L }[/math]. Similarly, an improper integral of a function, [math]\displaystyle{ \textstyle\int_0^\infty f(x)\,dx }[/math], is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if [math]\displaystyle{ \textstyle\int_0^\infty \left|f(x)\right|dx = L. }[/math]
- 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
- 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.^{[1]} 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.^{[2]}^{[3]}
- automatic differentiation
- In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation or computational differentiation,^{[4]}^{[5]} 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 [math]\displaystyle{ \tbinom{n}{k}. }[/math] It is the coefficient of the x^{k} term in the polynomial expansion of the binomial power (1 + x)^{n}, and it is given by the formula
- [math]\displaystyle{ \binom{n}{k} = \frac{n!}{k! (n-k)!}. }[/math]
- [math]\displaystyle{ |f(x)|\le M }[/math]
C
- calculus
- (From Latin calculus, literally 'small pebble', used for counting and calculations, as on an abacus)^{[6]} 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:^{[7]}
- 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:
- [math]\displaystyle{ (f\circ g)'=(f'\circ g)\cdot g'. }[/math]
- [math]\displaystyle{ F'(x) = f'(g(x)) g'(x). }[/math]
- [math]\displaystyle{ \frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx}. }[/math]
- [math]\displaystyle{ \frac{dz}{dx}=\frac{dz}{dy}\cdot\frac{dy}{dx} = f'(y)g'(x) = f'(g(x))g'(x). }[/math]
- 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.^{[8]} This definition typically applies to trigonometric functions.^{[9]}^{[10]} The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620).^{[11]}^{[12]}
- 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.^{[13]}^{[14]} This constant expresses an ambiguity inherent in the construction of antiderivatives. If a function [math]\displaystyle{ f(x) }[/math] is defined on an interval and [math]\displaystyle{ F(x) }[/math] is an antiderivative of [math]\displaystyle{ f(x) }[/math], then the set of all antiderivatives of [math]\displaystyle{ f(x) }[/math] is given by the functions [math]\displaystyle{ F(x) + C }[/math], where C is an arbitrary constant (meaning that any value for C makes [math]\displaystyle{ F(x) + C }[/math] 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.^{[15]}^{[16]}^{[17]}
- convergence tests
- Are methods of testing for the convergence, conditional convergence, absolute convergence, interval of convergence or divergence of an infinite series [math]\displaystyle{ \sum_{n=1}^\infty a_n }[/math].
- convergent series
- In mathematics, a series is the sum of the terms of an infinite sequence of numbers.
Given an infinite sequence [math]\displaystyle{ \left ( a_1,\ a_2,\ a_3,\dots \right ) }[/math], the nth partial sum [math]\displaystyle{ S_n }[/math] is the sum of the first n terms of the sequence, that is,
- [math]\displaystyle{ S_n = \sum_{k=1}^n a_k. }[/math]
- [math]\displaystyle{ \left | S_n - \ell \right \vert \le \ \varepsilon. }[/math]
- 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.^{[18]} Well-known examples of convex functions include the quadratic function [math]\displaystyle{ x^2 }[/math] and the exponential function [math]\displaystyle{ e^x }[/math].
- 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,^{[19]}^{[20]} although Colin Maclaurin also published special cases of the rule in 1748^{[21]} (and possibly knew of it as early as 1729).^{[22]}^{[23]}^{[24]}
- 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.^{[25]}^{[26]}
- 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.^{[27]}
- 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
- [math]\displaystyle{ dy = \frac{dy}{dx} \,dx, }[/math]
- differential calculus
- Is a subfield of calculus^{[28]} 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.^{[29]}
- 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
- [math]\displaystyle{ dy = f'(x)\,dx, }[/math]
- [math]\displaystyle{ dy = \frac{dy}{dx}\, dx }[/math]
- [math]\displaystyle{ df(x) = f'(x)\,dx. }[/math]
- 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.^{[30]} The test states that if [math]\displaystyle{ \{a_n\} }[/math] is a sequence of real numbers and [math]\displaystyle{ \{b_n\} }[/math] a sequence of complex numbers satisfying
- [math]\displaystyle{ a_{n+1} \le a_n }[/math]
- [math]\displaystyle{ \lim_{n \rightarrow \infty}a_n = 0 }[/math]
- [math]\displaystyle{ \left|\sum^{N}_{n=1}b_n\right|\leq M }[/math] for every positive integer N
- [math]\displaystyle{ \sum^{\infty}_{n=1}a_n b_n }[/math]
- 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
- 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 R^{2} are called double integrals, and integrals of a function of three variables over a region of R^{3} are called triple integrals.^{[31]}
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,^{[32]} 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^{[33]}
- [math]\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 }[/math]
- [math]\displaystyle{ f(x) = \int_{c}^{x} R \left(t, \sqrt{P(t)} \right) \, dt, }[/math]
- [math]\displaystyle{ f(x) = \begin{cases} \sin\frac{5}{x-1} & \mbox{ for } x \lt 1 \\ 0 & \mbox{ for } x = 1 \\ \frac{1}{x-1} & \mbox{ for } x \gt 1 \end{cases} }[/math]
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 [math]\displaystyle{ f(x)=ab^{cx+d} }[/math] is also an exponential function, as it can be rewritten as
- [math]\displaystyle{ ab^{cx+d}=\left(ab^d\right) \left(b^c\right)^x. }[/math]
- [math]\displaystyle{ f(c) \ge f(x) \ge f(d)\quad\text{for all }x\in [a,b]. }[/math]
- [math]\displaystyle{ m \lt f(x) \lt M\quad\text{for all }x \in [a,b]. }[/math]
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,^{[38]} considered the first published reference on the subject.^{[39]}
Perhaps the most well-known form of Faà di Bruno's formula says that
- [math]\displaystyle{ {d^n \over dx^n} f(g(x))=\sum \frac{n!}{m_1!\,1!^{m_1}\,m_2!\,2!^{m_2}\,\cdots\,m_n!\,n!^{m_n}}\cdot f^{(m_1+\cdots+m_n)}(g(x))\cdot \prod_{j=1}^n\left(g^{(j)}(x)\right)^{m_j}, }[/math]
- [math]\displaystyle{ 1\cdot m_1+2\cdot m_2+3\cdot m_3+\cdots+n\cdot m_n=n. }[/math]
- [math]\displaystyle{ {d^n \over dx^n} f(g(x)) =\sum \frac{n!}{m_1!\,m_2!\,\cdots\,m_n!}\cdot f^{(m_1+\cdots+m_n)}(g(x))\cdot \prod_{j=1}^n\left(\frac{g^{(j)}(x)}{j!}\right)^{m_j}. }[/math]
- [math]\displaystyle{ {d^n \over dx^n} f(g(x)) = \sum_{k=1}^n f^{(k)}(g(x))\cdot B_{n,k}\left(g'(x),g''(x),\dots,g^{(n-k+1)}(x)\right). }[/math]
- [math]\displaystyle{ D f(x) = \dfrac{d}{dx} f(x) }[/math],
- [math]\displaystyle{ J f(x) = \int_0^x\!\!\!\! f(s) {ds} }[/math],^{[Note 1]}
G
- general Leibniz rule
- The general Leibniz rule,^{[43]} named after Gottfried Wilhelm Leibniz, generalizes the product rule (which is also known as "Leibniz's rule"). It states that if [math]\displaystyle{ f }[/math] and [math]\displaystyle{ g }[/math] are [math]\displaystyle{ n }[/math]-times differentiable functions, then the product [math]\displaystyle{ fg }[/math] is also [math]\displaystyle{ n }[/math]-times differentiable and its [math]\displaystyle{ n }[/math]th derivative is given by
- [math]\displaystyle{ (fg)^{(n)}=\sum_{k=0}^n {n \choose k} f^{(n-k)} g^{(k)}, }[/math]
- global maximum
- 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).^{[44]}^{[45]}^{[46]} 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.
- global minimum
- 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).^{[47]}^{[48]}^{[49]} 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.
- golden spiral
- In geometry, a golden spiral is a logarithmic spiral whose growth factor is φ, the golden ratio.^{[50]} That is, a golden spiral gets wider (or further from its origin) by a factor of φ for every quarter turn it makes.
- gradient
- Is a multi-variable generalization of the derivative. While a derivative can be defined on functions of a single variable, for functions of several variables, the gradient takes its place. The gradient is a vector-valued function, as opposed to a derivative, which is scalar-valued.
H
- harmonic progression
- In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. It is a sequence of the form
- [math]\displaystyle{ \frac{1}{a} ,\ \frac{1}{a+d}\ , \frac{1}{a+2d}\ , \frac{1}{a+3d}\ , \cdots, \frac{1}{a+kd}, }[/math]
- higher derivative
- Let f be a differentiable function, and let f ′ be its derivative. The derivative of f ′ (if it has one) is written f ′′ and is called the second derivative of f. Similarly, the derivative of the second derivative, if it exists, is written f ′′′ and is called the third derivative of f. Continuing this process, one can define, if it exists, the nth derivative as the derivative of the (n-1)th derivative. These repeated derivatives are called higher-order derivatives. The nth derivative is also called the derivative of order n.
- homogeneous linear differential equation
- A differential equation can be homogeneous in either of two respects.
A first order differential equation is said to be homogeneous if it may be written
- [math]\displaystyle{ f(x,y)dy = g(x,y)dx, }[/math]
- [math]\displaystyle{ \frac{dx}x = h(u)du, }[/math]
- hyperbolic function
- Hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.
I
- identity function
- Also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument. In equations, the function is given by f(x) = x.
- imaginary number
- Is a complex number that can be written as a real number multiplied by the imaginary unit i,^{[note 1]} which is defined by its property i^{2} = −1.^{[52]} The square of an imaginary number bi is −b^{2}. For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary.^{[53]}
- implicit function
- In mathematics, an implicit equation is a relation of the form [math]\displaystyle{ R(x_1,\ldots, x_n) = 0 }[/math], where [math]\displaystyle{ R }[/math] is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is [math]\displaystyle{ x^2 +y^2-1 = 0 }[/math]. An implicit function is a function that is defined implicitly by an implicit equation, by associating one of the variables (the value) with the others (the arguments).^{[54]}^{:204–206} Thus, an implicit function for [math]\displaystyle{ y }[/math] in the context of the unit circle is defined implicitly by [math]\displaystyle{ x^2 +f(x)^2-1 = 0 }[/math]. This implicit equation defines [math]\displaystyle{ f }[/math] as a function of [math]\displaystyle{ x }[/math] only if [math]\displaystyle{ -1 \leq x \leq 1 }[/math] and one considers only non-negative (or non-positive) values for the values of the function. The implicit function theorem provides conditions under which some kinds of relations define an implicit function, namely relations defined as the indicator function of the zero set of some continuously differentiable multivariate function.
- improper fraction
- Common fractions can be classified as either proper or improper. When the numerator and the denominator are both positive, the fraction is called proper if the numerator is less than the denominator, and improper otherwise.^{[55]}^{[56]} In general, a common fraction is said to be a proper fraction if the absolute value of the fraction is strictly less than one—that is, if the fraction is greater than −1 and less than 1.^{[57]}^{[58]} It is said to be an improper fraction, or sometimes top-heavy fraction,^{[59]} if the absolute value of the fraction is greater than or equal to 1. Examples of proper fractions are 2/3, –3/4, and 4/9; examples of improper fractions are 9/4, –4/3, and 3/3.
- improper integral
- In mathematical analysis, an improper integral is the limit of a definite integral as an endpoint of the interval(s) of integration approaches either a specified real number, [math]\displaystyle{ \infty }[/math], [math]\displaystyle{ -\infty }[/math], or in some instances as both endpoints approach limits. Such an integral is often written symbolically just like a standard definite integral, in some cases with infinity as a limit of integration.
Specifically, an improper integral is a limit of the form:
- [math]\displaystyle{ \lim_{b\to\infty} \int_a^bf(x)\, dx, \qquad \lim_{a\to -\infty} \int_a^bf(x)\, dx, }[/math]
- [math]\displaystyle{ \lim_{c\to b^-} \int_a^cf(x)\, dx,\quad \lim_{c\to a^+} \int_c^bf(x)\, dx, }[/math]
- inflection point
- In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuous plane curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa.
- instantaneous rate of change
- The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable. .
- instantaneous velocity
- If we consider v as velocity and x as the displacement (change in position) vector, then we can express the (instantaneous) velocity of a particle or object, at any particular time t, as the derivative of the position with respect to time:
- [math]\displaystyle{ \boldsymbol{v} = \lim_{{\Delta t}\to 0} \frac{\Delta \boldsymbol{x}}{\Delta t} = \frac{d\boldsymbol{x}}{d\mathit{t}} . }[/math]
- [math]\displaystyle{ \boldsymbol{x} = \int \boldsymbol{v} \ d\mathit{t} . }[/math]
- integral
- An integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. .
- integral symbol
- The integral symbol: is used to denote integrals and antiderivatives in mathematics. .
- integrand
- The function to be integrated in an integral.
- integration by parts
- In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative. It is frequently used to transform the antiderivative of a product of functions into an antiderivative for which a solution can be more easily found. The rule can be readily derived by integrating the product rule of differentiation.
If u = u(x) and du = u′(x) dx, while v = v(x) and dv = v′(x) dx, then integration by parts states that:
- [math]\displaystyle{ \begin{align} \int_a^b u(x) v'(x) \, dx &= \Big[u(x) v(x)\Big]_a^b - \int_a^b u'(x) v(x) \, dx\\ &= u(b) v(b) - u(a) v(a) - \int_a^b u'(x) v(x) \, dx \end{align} }[/math]
- [math]\displaystyle{ \int u \, dv = uv - \int v \, du. }[/math]
- integration by substitution
- Also known as u-substitution, is a method for solving integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool in mathematics. It is the counterpart to the chain rule for differentiation. .
- intermediate value theorem
- In mathematical analysis, the intermediate value theorem states that if a continuous function, f, with an interval, [a, b], as its domain, takes values f(a) and f(b) at each end of the interval, then it also takes any value between f(a) and f(b) at some point within the interval.
This has two important corollaries:
- If a continuous function has values of opposite sign inside an interval, then it has a root in that interval (Bolzano's theorem).^{[62]}
- The image of a continuous function over an interval is itself an interval. .
- inverse trigonometric functions
- (Also called arcus functions,^{[63]}^{[64]}^{[65]}^{[66]}^{[67]} antitrigonometric functions^{[68]} or cyclometric functions^{[69]}^{[70]}^{[71]}) are the inverse functions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios.
J
- jump discontinuity
- Consider the function
- [math]\displaystyle{ f(x) = \begin{cases} x^2 & \mbox{ for } x \lt 1 \\ 0 & \mbox{ for } x = 1 \\ 2 - (x-1)^2 & \mbox{ for } x \gt 1 \end{cases} }[/math]
L
- Lebesgue integration
- In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the x-axis. The Lebesgue integral extends the integral to a larger class of functions. It also extends the domains on which these functions can be defined.
- L'Hôpital's rule
- L'Hôpital's rule or L'Hospital's rule uses derivatives to help evaluate limits involving indeterminate forms. Application (or repeated application) of the rule often converts an indeterminate form to an expression that can be evaluated by substitution, allowing easier evaluation of the limit. The rule is named after the 17th-century France mathematician Guillaume de l'Hôpital. Although the contribution of the rule is often attributed to L'Hôpital, the theorem was first introduced to L'Hôpital in 1694 by the Swiss mathematician Johann Bernoulli.
L'Hôpital's rule states that for functions f and g which are differentiable on an open interval I except possibly at a point c contained in I, if
[math]\displaystyle{ \lim_{x\to c}f(x)=\lim_{x\to c}g(x)=0 \text{ or } \pm\infty, }[/math] [math]\displaystyle{ g'(x)\ne 0 }[/math] for all x in I with x ≠ c, and [math]\displaystyle{ \lim_{x\to c}\frac{f'(x)}{g'(x)} }[/math] exists, then
- [math]\displaystyle{ \lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}. }[/math]
- limit comparison test
- The limit comparison test allows one to determine the convergence of one series based on the convergence of another.
- limit of a function
- .
- limits of integration
- .
- linear combination
- In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).^{[72]}^{[73]}^{[74]} The concept of linear combinations is central to linear algebra and related fields of mathematics.
- linear equation
- A linear equation is an equation relating two or more variables to each other in the form of [math]\displaystyle{ a_1x_1+\cdots +a_nx_n+b=0, }[/math] with the highest power of each variable being 1.
- linear system
- .
- list of integrals
- .
- logarithm
- .
- logarithmic differentiation
- .
- lower bound
- .
M
- mean value theorem
- .
- monotonic function
- .
- multiple integral
- .
- Multiplicative calculus
- .
- multivariable calculus
- .
N
- natural logarithm
- The natural logarithm of a number is its logarithm to the base of the mathematical constant e, where e is an irrational and transcendental number approximately equal to 2.718281828459. The natural logarithm of x is generally written as ln x, log_{e} x, or sometimes, if the base e is implicit, simply log x.^{[75]} Parentheses are sometimes added for clarity, giving ln(x), log_{e}(x) or log(x). This is done in particular when the argument to the logarithm is not a single symbol, to prevent ambiguity.
- non-Newtonian calculus
- .
- nonstandard calculus
- .
- notation for differentiation
- .
- numerical integration
- .
O
P
- Pappus's centroid theorem
- (Also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.
- parabola
- Is a plane curve that is mirror-symmetrical and is approximately U-shaped. It fits several superficially different other mathematical descriptions, which can all be proved to define exactly the same curves.
- paraboloid
- .
- partial derivative
- .
- partial differential equation
- .
- partial fraction decomposition
- .
- particular solution
- .
- piecewise-defined function
- A function defined by multiple sub-functions that apply to certain intervals of the function's domain.
- position vector
- .
- power rule
- .
- product integral
- .
- product rule
- .
- proper fraction
- .
- proper rational function
- .
- Pythagorean theorem
- .
- Pythagorean trigonometric identity
- .
Q
- quadratic function
- In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree. For example, a quadratic function in three variables x, y, and z contains exclusively terms x^{2}, y^{2}, z^{2}, xy, xz, yz, x, y, z, and a constant:
- [math]\displaystyle{ f(x,y,z)=ax^2+by^2+cz^2+dxy+exz+fyz+gx+hy+iz +j, }[/math]
- [math]\displaystyle{ f(x)=ax^2+bx+c,\quad a \ne 0 }[/math]
- [math]\displaystyle{ f(x,y) = a x^2 + by^2 + cx y+ d x+ ey + f \,\! }[/math]
- quadratic polynomial
- .
- quotient rule
- A formula for finding the derivative of a function that is the ratio of two functions.
R
- radian
- Is the SI unit for measuring angles, and is the standard unit of angular measure used in many areas of mathematics. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends; one radian is just under 57.3 degrees (expansion at OEIS: A072097). The unit was formerly an SI supplementary unit, but this category was abolished in 1995 and the radian is now considered an SI derived unit.^{[77]} Separately, the SI unit of solid angle measurement is the steradian .
- ratio test
- .
- reciprocal function
- .
- reciprocal rule
- .
- Riemann integral
- .
- .
- removable discontinuity
- .
- Rolle's theorem
- .
- root test
- .
S
- scalar
- .
- secant line
- .
- second-degree polynomial
- .
- second derivative
- .
- second derivative test
- .
- second-order differential equation
- .
- series
- .
- shell integration
- .
- Simpson's rule
- .
- sine
- .
- sine wave
- .
- slope field
- .
- squeeze theorem
- .
- sum rule in differentiation
- .
- sum rule in integration
- .
- summation
- .
- supplementary angle
- .
- surface area
- .
- system of linear equations
- .
T
- table of integrals
- .
- Taylor series
- .
- Taylor's theorem
- .
- tangent
- .
- third-degree polynomial
- .
- third derivative
- .
- toroid
- .
- total differential
- .
- trigonometric functions
- .
- trigonometric identities
- .
- trigonometric integral
- .
- trigonometric substitution
- .
- trigonometry
- .
- triple integral
- .
U
V
- variable
- .
- vector
- .
- vector calculus
- .
W
- washer
- .
- washer method
- .
See also
- Outline of calculus
- Glossary of areas of mathematics
- Glossary of astronomy
- Glossary of biology
- Glossary of botany
- Glossary of chemistry
- Glossary of ecology
- Glossary of engineering
- Glossary of physics
- Glossary of probability and statistics
References
- ↑ "Asymptotes" by Louis A. Talman
- ↑ Williamson, Benjamin (1899), "Asymptotes", An elementary treatise on the differential calculus, https://books.google.com/books?id=znsXAAAAYAAJ&pg=241
- ↑ Nunemacher, Jeffrey (1999), "Asymptotes, Cubic Curves, and the Projective Plane", Mathematics Magazine 72 (3): 183–192, doi:10.2307/2690881
- ↑ Neidinger, Richard D. (2010). "Introduction to Automatic Differentiation and MATLAB Object-Oriented Programming". SIAM Review 52 (3): 545–563. doi:10.1137/080743627. http://academics.davidson.edu/math/neidinger/SIAMRev74362.pdf.
- ↑ Baydin, Atilim Gunes; Pearlmutter, Barak; Radul, Alexey Andreyevich; Siskind, Jeffrey (2018). "Automatic differentiation in machine learning: a survey". Journal of Machine Learning Research 18: 1–43. http://jmlr.org/papers/v18/17-468.html.
- ↑ "Calculus". http://www.oxforddictionaries.com/us/definition/american_english/calculus.
- ↑ Howard Eves, "Two Surprising Theorems on Cavalieri Congruence", The College Mathematics Journal, volume 22, number 2, March, 1991), pages 118–124
- ↑ "Chapter II. The Acute Angle [10 Functions of complementary angles"]. written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. January 1909. pp. 11–12. https://archive.org/stream/planetrigonometr00hallrich#page/n26/mode/1up. Retrieved 2017-08-12.
- ↑ Algebra and Trigonometry (8 ed.). Cengage Learning. 2014. p. 528. ISBN 978-128596583-3. https://books.google.com/books?id=JEDAAgAAQBAJ&pg=PA528. Retrieved 2017-07-28.
- ↑ "5.1 The Elementary Identities". Precalculus. 2012. http://jwbales.home.mindspring.com/precal/part5/part5.1.html.
- ↑ Canon triangulorum. 1620.
- ↑ Roegel, Denis, ed (2010-12-06). "A reconstruction of Gunter's Canon triangulorum (1620)". HAL. https://hal.inria.fr/inria-00543938/document.
- ↑ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8. https://archive.org/details/calculusearlytra00stew_1.
- ↑ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 978-0-547-16702-2.
- ↑ Stalker, John (1998). Complex Analysis: Fundamentals of the Classical Theory of Functions. Springer. p. 77. ISBN 0-8176-4038-X. https://books.google.com/books?id=yl3GIXd3dFIC&q=%22calculus+of+residues%22&pg=PP12.
- ↑ Bak, Joseph; Newman, Donald J. (1997). "Chapters 11 & 12". Complex Analysis. Springer. pp. 130–156. ISBN 0-387-94756-6. https://books.google.com/books?id=JX2YSgfZwbYC&q=%22contour+integral%22&pg=PA130.
- ↑ Krantz, Steven George (1999). "Chapter 2". Handbook of Complex Variables. Springer. ISBN 0-8176-4011-8. https://books.google.com/books?id=aYU2AdF_0dIC&q=Calculus++Residues+inauthor:krantz&pg=PT13.
- ↑ "Lecture Notes 2". http://www.stat.cmu.edu/~larry/=stat705/Lecture2.pdf.
- ↑ Cramer, Gabriel (1750). "Introduction à l'Analyse des lignes Courbes algébriques" (in fr). Geneva: Europeana. pp. 656–659. https://www.europeana.eu/resolve/record/03486/E71FE3799CEC1F8E2B76962513829D2E36B63015.
- ↑ Kosinski, A. A. (2001). "Cramer's Rule is due to Cramer". Mathematics Magazine 74 (4): 310–312. doi:10.2307/2691101.
- ↑ MacLaurin, Colin (1748). A Treatise of Algebra, in Three Parts.. Printed for A. Millar & J. Nourse. https://archive.org/details/atreatisealgebr03maclgoog.
- ↑ Boyer, Carl B. (1968). A History of Mathematics (2nd ed.). Wiley. pp. 431.
- ↑ Katz, Victor (2004). A History of Mathematics (Brief ed.). Pearson Education. pp. 378–379.
- ↑ Hedman, Bruce A. (1999). "An Earlier Date for "Cramer's Rule"". Historia Mathematica 26 (4): 365–368. doi:10.1006/hmat.1999.2247. http://professorhedman.com/Cramers.Rule.pdf.
- ↑ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8. https://archive.org/details/calculusearlytra00stew_1.
- ↑ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 978-0-547-16702-2.
- ↑ Douglas C. Giancoli (2000). [Physics for Scientists and Engineers with Modern Physics (3rd Edition)]. Prentice Hall. ISBN:0-13-021517-1
- ↑ "Definition of DIFFERENTIAL CALCULUS" (in en). https://www.merriam-webster.com/dictionary/differential%20calculus.
- ↑ "Integral Calculus - Definition of Integral calculus by Merriam-Webster" (in en). https://www.merriam-webster.com/dictionary/integral%20calculus.
- ↑ Démonstration d’un théorème d’Abel. Journal de mathématiques pures et appliquées 2nd series, tome 7 (1862), p. 253-255 .
- ↑ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks Cole Cengage Learning. ISBN 978-0-495-01166-8. https://archive.org/details/calculusearlytra00stew_1.
- ↑ Oxford English Dictionary, 2nd ed.: natural logarithm
- ↑ Encyclopedic Dictionary of Mathematics 142.D
- ↑ Butcher 2003, p. 45; Hairer, Nørsett & Wanner 1993, p. 35
- ↑ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8. https://archive.org/details/calculusearlytra00stew_1.
- ↑ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 978-0-547-16702-2.
- ↑ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 978-0-321-58876-0.
- ↑ (Arbogast 1800).
- ↑ According to (Craik 2005): see also the analysis of Arbogast's work by (Johnson 2002).
- ↑ William F. Kern, James R. Bland, Solid Mensuration with proofs, 1938, p. 67
- ↑ MacLane, Saunders; Birkhoff, Garrett (1967). Algebra (First ed.). New York: Macmillan. pp. 1–13. https://archive.org/details/algebra00macl.
- ↑ Spivak, Michael (1980), Calculus (2nd ed.), Houston, Texas: Publish or Perish Inc.
- ↑ Olver, Peter J. (2000). Applications of Lie Groups to Differential Equations. Springer. pp. 318–319. ISBN 9780387950006. https://books.google.com/books?id=sI2bAxgLMXYC&pg=PA318.
- ↑ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8. https://archive.org/details/calculusearlytra00stew_1.
- ↑ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 978-0-547-16702-2.
- ↑ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 978-0-321-58876-0.
- ↑ Stewart, James (2008). Calculus: Early Transcendentals (6th ed.). Brooks/Cole. ISBN 978-0-495-01166-8. https://archive.org/details/calculusearlytra00stew_1.
- ↑ Larson, Ron; Edwards, Bruce H. (2009). Calculus (9th ed.). Brooks/Cole. ISBN 978-0-547-16702-2.
- ↑ Thomas, George B.; Weir, Maurice D.; Hass, Joel (2010). Thomas' Calculus: Early Transcendentals (12th ed.). Addison-Wesley. ISBN 978-0-321-58876-0.
- ↑ Chang, Yu-sung, "Golden Spiral ", The Wolfram Demonstrations Project.
- ↑ Erdős, P. (1932), "Egy Kürschák-féle elemi számelméleti tétel általánosítása" (in hu), Mat. Fiz. Lapok 39: 17–24, https://www.renyi.hu/~p_erdos/1932-02.pdf. As cited by "Paul Erdős and Egyptian fractions", Erdős centennial, Bolyai Soc. Math. Stud., 25, János Bolyai Math. Soc., Budapest, 2013, pp. 289–309, doi:10.1007/978-3-642-39286-3_9.
- ↑ Uno Ingard, K. (1988). "Chapter 2". Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X. https://books.google.com/books?id=SGVfGIewvxkC&pg=PA38.
- ↑ Sinha, K.C. (2008). A Text Book of Mathematics Class XI (Second ed.). Rastogi Publications. p. 11.2. ISBN 978-81-7133-912-9. https://books.google.com/books?id=mqdzqbPYiAUC&pg=SA11-PA2.
- ↑ Chiang, Alpha C. (1984). Fundamental Methods of Mathematical Economics (Third ed.). New York: McGraw-Hill. ISBN 0-07-010813-7. https://books.google.com/books?id=6gcoAQAAMAAJ.
- ↑ "World Wide Words: Vulgar fractions". World Wide Words. http://www.worldwidewords.org/qa/qa-vul1.htm.
- ↑ Weisstein, Eric W.. "Improper Fraction". http://mathworld.wolfram.com/ImproperFraction.html.
- ↑ Laurel (31 March 2004). "Math Forum – Ask Dr. Math:Can Negative Fractions Also Be Proper or Improper?". http://mathforum.org/library/drmath/view/65128.html.
- ↑ "New England Compact Math Resources". http://www.necompact.org/ea/gle_support/Math/resources_number/prop_fraction.htm.
- ↑ Greer, A. (1986). New comprehensive mathematics for 'O' level (2nd ed., reprinted. ed.). Cheltenham: Thornes. p. 5. ISBN 978-0-85950-159-0. https://books.google.com/books?id=wX2dxeDahAwC&pg=PA5. Retrieved 2014-07-29.
- ↑ "Brook Taylor". History.MCS.St-Andrews.ac.uk. http://www-history.mcs.st-andrews.ac.uk/Biographies/Taylor.html.
- ↑ "Brook Taylor". Stetson.edu. https://www2.stetson.edu/~efriedma/periodictable/html/Tl.html.
- ↑ Weisstein, Eric W.. "Bolzano's Theorem". http://mathworld.wolfram.com/BolzanosTheorem.html.
- ↑ Taczanowski, Stefan (1978-10-01). "On the optimization of some geometric parameters in 14 MeV neutron activation analysis". Nuclear Instruments and Methods. ScienceDirect. 155(3): 543–546. doi:10.1016/0029-554X(78)90541-4.
- ↑ Hazewinkel, Michiel (1994) [1987]. Encyclopaedia of Mathematics (unabridged reprint ed.). Kluwer Academic Publishers / Springer Science & Business Media. ISBN:978-155608010-4.
- ↑ Ebner, Dieter (2005-07-25). Preparatory Course in Mathematics (PDF) (6 ed.). Department of Physics, University of Konstanz. Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26.
- ↑ Mejlbro, Leif (2010-11-11). Stability, Riemann Surfaces, Conformal Mappings - Complex Functions Theory (PDF) (1 ed.). Ventus Publishing ApS / Bookboon. ISBN:978-87-7681-702-2. Archived (PDF) from the original on 2017-07-26. Retrieved 2017-07-26.
- ↑ Durán, Mario (2012). Mathematical methods for wave propagation in science and engineering. 1: Fundamentals (1 ed.). Ediciones UC. p. 88. ISBN:978-956141314-6.
- ↑ Hall, Arthur Graham; Frink, Fred Goodrich (January 1909). "Chapter II. The Acute Angle [14] Inverse trigonometric functions". Written at Ann Arbor, Michigan, USA. Trigonometry. Part I: Plane Trigonometry. New York, USA: Henry Holt and Company / Norwood Press / J. S. Cushing Co. - Berwick & Smith Co., Norwood, Massachusetts, USA. p. 15. Retrieved 2017-08-12. […] α = arcsin m: It is frequently read "arc-sinem" or "anti-sine m," since two mutually inverse functions are said each to be the anti-function of the other. […] A similar symbolic relation holds for the other trigonometric functions. […] This notation is universally used in Europe and is fast gaining ground in this country. A less desirable symbol, α = sin-1m, is still found in English and American texts. The notation α = inv sin m is perhaps better still on account of its general applicability. […]
- ↑ Klein, Christian Felix (1924) [1902]. Elementarmathematik vom höheren Standpunkt aus: Arithmetik, Algebra, Analysis (in German). 1 (3rd ed.). Berlin: J. Springer.
- ↑ Klein, Christian Felix (2004) [1932]. Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis. Translated by Hedrick, E. R.; Noble, C. A. (Translation of 3rd German ed.). Dover Publications, Inc. / The Macmillan Company. ISBN:978-0-48643480-3. Retrieved 2017-08-13.
- ↑ Dörrie, Heinrich (1965). Triumph der Mathematik. Translated by Antin, David. Dover Publications. p. 69. ISBN:978-0-486-61348-2.
- ↑ Lay, David C. (2006). Linear Algebra and Its Applications (3rd ed.). Addison–Wesley. ISBN 0-321-28713-4. https://archive.org/details/studyguidetoline0000layd.
- ↑ Strang, Gilbert (2006). Linear Algebra and Its Applications (4th ed.). Brooks Cole. ISBN 0-03-010567-6.
- ↑ Axler, Sheldon (2002). Linear Algebra Done Right (2nd ed.). Springer. ISBN 0-387-98258-2.
- ↑ Mathematics for physical chemistry (3rd ed.). Academic Press. 2005. p. 9. ISBN 0-12-508347-5. https://books.google.com/books?id=nGoSv5tmATsC. Extract of page 9
- ↑ "Quadratic Equation -- from Wolfram MathWorld". http://mathworld.wolfram.com/QuadraticEquation.html.
- ↑ "Resolution 8 of the CGPM at its 20th Meeting (1995)". Bureau International des Poids et Mesures. http://www.bipm.org/en/CGPM/db/20/8/.
Works cited
- Apostol, T (1967), Calculus, Vol. 1 (2nd ed.), Jon Wiley & Sons.
- Arbogast, L. F. A. (1800) (in fr), Du calcul des derivations, Strasbourg: Levrault, pp. xxiii+404, https://books.google.com/books?id=YoPq8uCy5Y8C.
- Butcher, John C. (2003). Numerical Methods for Ordinary Differential Equations. New York: John Wiley & Sons. ISBN 978-0-471-96758-3.
- Craik, Alex D. D. (February 2005), "Prehistory of Faà di Bruno's Formula", American Mathematical Monthly 112 (2): 217–234, doi:10.2307/30037410.
- Hairer, Ernst; Nørsett, Syvert Paul; Wanner, Gerhard (1993). Solving ordinary differential equations I: Nonstiff problems. Berlin, New York: Springer-Verlag. ISBN 978-3-540-56670-0.
- Johnson, Warren P. (March 2002), "The Curious History of Faà di Bruno's Formula", American Mathematical Monthly 109 (3): 217–234, doi:10.2307/2695352, https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Johnson217-234.pdf.
Notes
- ↑ j is usually used in Engineering contexts where i has other meanings (such as electrical current)
- ↑ The symbol J is commonly used instead of the intuitive I in order to avoid confusion with other concepts identified by similar I–like glyphs, e.g. identities.
Original source: https://en.wikipedia.org/wiki/Glossary of calculus.
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