Functional derivative

In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional 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:

The coefficient of δf(x), denoted as δJ/δf(x), is called the functional derivative of J with respect to f at the point x. For this example functional, the functional derivative is the left hand side of the Euler-Lagrange equation,

In this section, the functional derivative is defined. Then the functional differential is defined in terms of the functional derivative.

Given a manifold M representing (continuous/smooth) functions ρ (with certain boundary conditions etc.), and a functional F defined as

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