Functional derivative

In mathematics and theoretical physics, 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.

Two possible, restricted definitions suitable for certain computations are given here. There are more general definitions of functional derivatives.

For any functional F mapping (continuous/smooth/with certain boundary conditions/etc.) functions φ from a manifold M to <math>\mathbb{R}<math> or <math>\mathbb{C}<math>, then, provided the following derivative exists, the functional derivative

<math>\frac{\delta F}{\delta \phi}[\phi]<math>

is a distribution such that for all test functions f,

<math>\left(\frac{\delta F}{\delta

\phi}[\phi]\right)[f]=\frac{d}{d\epsilon}F[\phi+\epsilon f].<math>

Another definition is in terms of a limit and the Dirac delta function, δ:

<math>\frac{\delta F[\phi(x)]}{\delta \phi(y)}=\lim_{\varepsilon\to 0}\frac{F[\phi(x)+\varepsilon\delta(x-y)]-F[\phi(y)]}{\varepsilon}.

<math>