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 a functional F mapping (continuous/smooth/with certain boundary conditions/etc.) functions φ from a manifold M to R or C, the functional derivative of F, denoted δF is a distribution such that for all test functions f,

[itex]\delta F[f]=\frac{d}{d\epsilon}F[\phi+\epsilon f].[itex]

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

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

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