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,
![\delta F[f]=\frac{d}{d\epsilon}F[\phi+\epsilon f].](/img/math/8cf379e21c356666827006ad511f219c.png)
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}.](/img/math/86760a7085df0dd8bd482e6d99228a42.png)