# 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).