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Concept
Functional derivative
Summary
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 acts on functions) to a change in a function on which the functional depends.
In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integrand L of a functional, if a function f is varied by adding to it another function δf that is arbitrarily small, and the resulting integrand is expanded in powers of δf, the coefficient of δf in the first order term is called the functional derivative.
For example, consider the functional
where f ′(x) ≡ df/dx. If f is varied by adding to it a function δf, and the resulting integrand L(x, f +δf, f '+δf ′) is expanded in powers of δf, then the change in the value of J to first order in δf can be expressed as follows:
where the variation in the derivative, δf ′ was rewritten as the derivative of the variation (δf) ′, and integration by parts was used in these derivatives.
In this section, the functional differential (or variation or first variation) Called first variation in , variation or first variation in , variation or differential in and differential in . is defined. Then the functional derivative is defined in terms of the functional differential.
Suppose is a Banach space and is a functional defined on .
The differential of at a point is the linear functional on defined by the condition that, for all ,
where is a real number that depends on in such a way that as . This means that is the Fréchet derivative of at .
However, this notion of functional differential is so strong it may not exist, and in those cases a weaker notion, like the Gateaux derivative is preferred. In many practical cases, the functional differential is defined as the directional derivative
Note that this notion of the functional differential can even be defined without a norm.
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