Functional derivative

Volterra derivative

One of the first concepts of a derivative in an infinite-dimensional space. Let be some functional of a continuous function of one variable ; let be some interior point of the segment ; let , where the variation is different from zero in a small neighbourhood of ; and let . The limit

assuming that it exists, is called the functional derivative of and is denoted by . For example, for the simplest functional of the classical calculus of variations,

the functional derivative has the form

that is, it is the left-hand side of the Euler equation, which is a necessary condition for a minimum of .

In theoretical questions the concept of a functional derivative has only historical interest, and in practice has been supplanted by the concepts of the Gâteaux derivative and the Fréchet derivative. But the concept of a functional derivative has been applied with success in numerical methods of the classical calculus of variations (see Variational calculus, numerical methods of).

Comments

The existence of the functional derivative of at and apparently means that the Fréchet derivative of at , which is a continuous linear form on the space of admissible infinitesimal variations , is of the form for some continuous function , so that it can be continuously extended to the -function at . In the example this happens only if is twice continuously differentiable.