Category: Differential calculus

Automatic differentiation
In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation, computational differentiation, auto-differentiation, or simply autodiff, is a set of techn
Infinitely near point
In algebraic geometry, an infinitely near point of an algebraic surface S is a point on a surface obtained from S by repeatedly blowing up points. Infinitely near points of algebraic surfaces were int
deal.II is a free, open-source library to solve partial differential equations using the finite element method. The current release is version 9.4, released in June 2022. It is one of the most widely
Differentiable vector–valued functions from Euclidean space
In the mathematical discipline of functional analysis, a differentiable vector-valued function from Euclidean space is a differentiable function valued in a topological vector space (TVS) whose domain
Fluent (mathematics)
A fluent is a time-varying quantity or variable. The term was used by Isaac Newton in his early calculus to describe his form of a function. The concept was introduced by Newton in 1665 and detailed i
Jacobian matrix and determinant
In vector calculus, the Jacobian matrix (/dʒəˈkoʊbiən/, /dʒɪ-, jɪ-/) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. When this matrix is squa
Method of Fluxions
Method of Fluxions (Latin: De Methodis Serierum et Fluxionum) is a mathematical treatise by Sir Isaac Newton which served as the earliest written formulation of modern calculus. The book was completed
Parametric derivative
In calculus, a parametric derivative is a derivative of a dependent variable with respect to another dependent variable that is taken when both variables depend on an independent third variable, usual
Second derivative
In calculus, the second derivative, or the second order derivative, of a function f is the derivative of the derivative of f. Roughly speaking, the second derivative measures how the rate of change of
Checkpointing scheme
Checkpointing schemes are scientific computing algorithms used in solving time dependent adjoint equations, as well as reverse mode automatic differentiation.
Symmetric derivative
In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as The expression under the limit is sometimes called the symmetric difference quotient. A
In mathematics, linearization is finding the linear approximation to a function at a given point. The linear approximation of a function is the first order Taylor expansion around the point of interes
Difference quotient
In single-variable calculus, the difference quotient is usually the name for the expression which when taken to the limit as h approaches 0 gives the derivative of the function f. The name of the expr
In mathematics, 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). Derivati
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 where is the der
In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) whose value at a point is the "direction and rate of
Inflection point
In differential calculus and differential geometry, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a smooth plane curve at which the curvature
Third derivative
In calculus, a branch of mathematics, the third derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing. The third derivative of a function can
Differential calculus
In mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. It is one of the two traditional divisions of calculus, the other being integral calc
Differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-verti
Flat function
In mathematics, especially real analysis, a flat function is a smooth function all of whose derivatives vanish at a given point . The flat functions are, in some sense, the antitheses of the analytic
Numerical differentiation
In numerical analysis, numerical differentiation algorithms estimate the derivative of a mathematical function or function subroutine using values of the function and perhaps other knowledge about the
Reduced derivative
In mathematics, the reduced derivative is a generalization of the notion of derivative that is well-suited to the study of functions of bounded variation. Although functions of bounded variation have
Notation for differentiation
In differential calculus, there is no single uniform notation for differentiation. Instead, various notations for the derivative of a function or variable have been proposed by various mathematicians.
Laplace transform applied to differential equations
In mathematics, the Laplace transform is a powerful integral transform used to switch a function from the time domain to the s-domain. The Laplace transform can be used in some cases to solve linear d
Differential coefficient
In physics, the differential coefficient of a function f(x) is what is now called its derivative df(x)/dx, the (not necessarily constant) multiplicative factor or coefficient of the differential dx in
Fermat's theorem (stationary points)
In mathematics, Fermat's theorem (also known as interior extremum theorem) is a method to find local maxima and minima of differentiable functions on open sets by showing that every local extremum of
Skew gradient
In mathematics, a skew gradient of a harmonic function over a simply connected domain with two real dimensions is a vector field that is everywhere orthogonal to the gradient of the function and that
Crinkled arc
In mathematics, and in particular the study of Hilbert spaces, a crinkled arc is a type of continuous curve. The concept is usually credited to Paul Halmos. Specifically, consider where is a Hilbert s
Fundamental matrix (linear differential equation)
In mathematics, a fundamental matrix of a system of n homogeneous linear ordinary differential equations is a matrix-valued function whose columns are linearly independent solutions of the system.Then
Racetrack principle
In calculus, the racetrack principle describes the movement and growth of two functions in terms of their derivatives. This principle is derived from the fact that if a horse named Frank Fleetfeet alw
Institutiones calculi differentialis
Institutiones calculi differentialis (Foundations of differential calculus) is a mathematical work written in 1748 by Leonhard Euler and published in 1755 that lays the groundwork for the differential
Differentiation rules
This is a summary of differentiation rules, that is, rules for computing the derivative of a function in calculus.
Adjoint equation
An adjoint equation is a linear differential equation, usually derived from its primal equation using integration by parts. Gradient values with respect to a particular quantity of interest can be eff
Implicit function
In mathematics, an implicit equation is a relation of the form where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is An implicit fun
Directional derivative
In mathematics, the directional derivative of a multivariable differentiable (scalar) function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the
Total derivative
In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total de
In mathematics, in the area of combinatorics and quantum calculus, the q-derivative, or Jackson derivative, is a q-analog of the ordinary derivative, introduced by Frank Hilton Jackson. It is the inve
Leibniz integral rule
In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form where and the integral are functions dependen
Metric derivative
In mathematics, the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces. It generalizes the notion of "speed" or "absolute velocity" to spaces which have a n
Stationary point
In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. Informally, i
Logarithmic differentiation
In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f, The technique is o
Differential calculus over commutative algebras
In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formul
Time evolution of integrals
Within differential calculus, in many applications, one needs to calculate the rate of change of a volume or surface integral whose domain of integration, as well as the integrand, are functions of a
Convenient vector space
In mathematics, convenient vector spaces are locally convex vector spaces satisfying a very mild completeness condition. Traditional differential calculus is effective in the analysis of finite-dimens
Differentiation of trigonometric functions
The differentiation of trigonometric functions is the mathematical process of finding the derivative of a trigonometric function, or its rate of change with respect to a variable. For example, the der
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after Francesco Faà di Bruno , although he was not the first to state or prove the f
Functional derivative
In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that a
Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to
Straightening theorem for vector fields
In differential calculus, the domain-straightening theorem states that, given a vector field on a manifold, there exist local coordinates such that in a neighborhood of a point where is nonzero. The t
Ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those function
Differentiable programming
Differentiable programming is a programming paradigm in which a numeric computer program can be differentiated throughout via automatic differentiation. This allows for gradient-based optimization of
In calculus, a branch of mathematics, the notions of one-sided differentiability and semi-differentiability of a real-valued function f of a real variable are weaker than differentiability. Specifical
Symmetrically continuous function
In mathematics, a function is symmetrically continuous at a point x if The usual definition of continuity implies symmetric continuity, but the converse is not true. For example, the function is symme
Binomial differential equation
In mathematics, the binomial differential equation is an ordinary differential equation containing one or more functions of one independent variable and the derivatives of those functions. For example
Leibniz's notation
In calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent infinitely small (or infinit
Quantum calculus
Quantum calculus, sometimes called calculus without limits, is equivalent to traditional infinitesimal calculus without the notion of limits. It defines "q-calculus" and "h-calculus", where h ostensib
Euler–Poisson–Darboux equation
In mathematics, the Euler–Poisson–Darboux equation is the partial differential equation This equation is named for Siméon Poisson, Leonhard Euler, and Gaston Darboux. It plays an important role in sol
Infinite-dimensional vector function
An infinite-dimensional vector function is a function whose values lie in an infinite-dimensional topological vector space, such as a Hilbert space or a Banach space. Such functions are applied in mos
Time derivative
A time derivative is a derivative of a function with respect to time, usually interpreted as the rate of change of the value of the function. The variable denoting time is usually written as .
Linearity of differentiation
In calculus, the derivative of any linear combination of functions equals the same linear combination of the derivatives of the functions; this property is known as linearity of differentiation, the r
Derivative test
In calculus, 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. D
Hyperbolic angle
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit
Differential-algebraic system of equations
In mathematics, a differential-algebraic system of equations (DAEs) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system. Suc
Differential (mathematics)
In mathematics, differential refers to several related notions derived from the early days of calculus, put on a rigorous footing, such as infinitesimal differences and the derivatives of functions. T
Logarithmic derivative
In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula where is the derivative of f. Intuitively, this is the infinitesimal
Darboux derivative
The Darboux derivative of a map between a manifold and a Lie group is a variant of the standard derivative. It is arguably a more natural generalization of the single-variable derivative. It allows a
A fluxion is the instantaneous rate of change, or gradient, of a fluent (a time-varying quantity, or function) at a given point. Fluxions were introduced by Isaac Newton to describe his form of a time
Fundamental increment lemma
In single-variable differential calculus, the fundamental increment lemma is an immediate consequence of the definition of the derivative f'(a) of a function f at a point a: The lemma asserts that the
Related rates
In differential calculus, related rates problems involve finding a rate at which a quantity changes by relating that quantity to other quantities whose rates of change are known. The rate of change is
Hessian automatic differentiation
In applied mathematics, Hessian automatic differentiation are techniques based on automatic differentiation (AD)that calculate the second derivative of an -dimensional function, known as the Hessian m
Differentiation in Fréchet spaces
In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is
Linear approximation
In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function). They are widely used in the method of finite differences
Inexact differential equation
An inexact differential equation is a differential equation of the form (see also: inexact differential) The solution to such equations came with the invention of the integrating factor by Leonhard Eu