In the
calculus of variations
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
, a field of
mathematical analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (m ...
, the functional derivative (or variational derivative)
relates a change in a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
(a functional in this sense is a function that acts on functions) to a change in a
function
Function or functionality may refer to:
Computing
* Function key, a type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
on which the functional depends.
In the calculus of variations, functionals are usually expressed in terms of an
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
of functions, their
arguments
An argument is a statement or group of statements called premises intended to determine the degree of truth or acceptability of another statement called conclusion. Arguments can be studied from three main perspectives: the logical, the dialectic ...
, and their
derivative
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). Derivatives are a fundamental tool of calculus. F ...
s. In an integral of a functional, if a function is varied by adding to it another function that is arbitrarily small, and the resulting integrand is expanded in powers of , the coefficient of in the first order term is called the functional derivative.
For example, consider the functional
where . If is varied by adding to it a function , and the resulting integrand is expanded in powers of , then the change in the value of to first order in can be expressed as follows:
[According to , this notation is customary in ]physical
Physical may refer to:
*Physical examination
In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally co ...
literature.
where the variation in the derivative, was rewritten as the derivative of the variation , and
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 the product of their derivative and antiderivative. ...
was used.
Definition
In this section, the functional derivative is defined. Then the functional differential is defined in terms of the functional derivative.
Functional derivative
Given a
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
representing (
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
/
smooth
Smooth may refer to:
Mathematics
* Smooth function, a function that is infinitely differentiable; used in calculus and topology
* Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
* Smooth algebrai ...
) functions (with certain
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
s etc.), and a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
defined as
the functional derivative of , denoted , is defined through
[.]
where
is an arbitrary function. The quantity
is called the variation of .
In other words,
is a linear functional, so one may apply the Riesz–Markov–Kakutani representation theorem to represent this functional as integration against some measure (mathematics), measure.
Then is defined to be the Radon–Nikodym derivative of this measure.
One thinks of the function as the gradient of at the point (that is, how much the functional will change if the function is changed at the point ) and
as the directional derivative at point in the direction of . Then analogous to vector calculus, the inner product with the gradient gives the directional derivative.
Functional differential
The differential (or variation or first variation) of the functional