TOPICS
Search

Functional Derivative


The functional derivative is a generalization of the usual derivative that arises in the calculus of variations. In a functional derivative, instead of differentiating a function with respect to a variable, one differentiates a functional with respect to a function. The definition for the univariate case is

 (deltaF[f(x)])/(deltaf(y))=lim_(epsilon->0)(F[f(x)+epsilondelta(x-y)]-F[f(x)])/epsilon.

For example, the Euler-Lagrange differential equation is the result of functional differentiation of the Hamiltonian action (functional).


See also

Calculus of Variations, Derivative, Euler-Lagrange Differential Equation, Euler-Lagrange Derivative, Functional, Variation

This entry contributed by Michael Trott

Explore with Wolfram|Alpha

Cite this as:

Trott, Michael. "Functional Derivative." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FunctionalDerivative.html

Subject classifications