Functional derivative

From Exampleproblems

In mathematics and theoretical physics, the functional derivative is a generalization of the directional derivative. The difference is that for the former one differentiates in the direction of a vector, while for the latter in the direction of a function. Both of these can be viewed as extensions of the usual calculus derivative.

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,

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

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

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

Formal description

The definition of a functional derivative may be made much more mathematically precise and formal by defining the space of functions more carefully. For example, when the space of functions is a Banach space, the functional derivative becomes known as the Gâteaux derivative, and on Fréchet spaces, the Fréchet derivative. Note that the well-known Hilbert space is a special case of a Banach space. The more formal treatment allows many theorems from ordinary calculus and analysis to be generalized to corresponding theorems in functional analysis, as well as numerous new theorems to be stated.es:Derivada funcional