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.

If we have a functional F mapping (continuous/smooth/with certain boundary conditions/etc.) functions $\phi$ from a manifold M to $\mathbb {R}$ or $\mathbb {C}$, 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,

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

Another definition is in terms of a limit and the Dirac delta function, $\delta (x)$:

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