# 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.

**How to Cite This Entry:**

Functional derivative.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Functional_derivative&oldid=34863