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# 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 $\mathbb{R}$ or $\mathbb{C}$, then, provided the following derivative exists, the functional derivative

$\frac{\delta F}{\delta \phi}[\phi]$

is a distribution such that for all test functions f,

$\left(\frac{\delta F}{\delta \phi}[\phi]\right)[f]=\frac{d}{d\epsilon}F[\phi+\epsilon f].$

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

$\frac{\delta F[\phi(x)]}{\delta \phi(y)}=\lim_{\varepsilon\to 0}\frac{F[\phi(x)+\varepsilon\delta(x-y)]-F[\phi(y)]}{\varepsilon}.$
03-10-2013 05:06:04