Functional Derivative

the functional derivative of F, denoted, is a distribution such that for all test functions f

$\begin{align} \left\langle \frac{\delta F}{\delta\varphi(x)}, f(x) \right\rangle &= \int \frac{\delta F}{\delta\varphi(x')} f(x')dx' \\ &= \lim_{\varepsilon\to 0}\frac{F-F}{\varepsilon} \\ &= \left.\frac{d}{d\epsilon}F\right|_{\epsilon=0}. \end{align}$

Using the first variation of, in place of yields the first variation of, ; this is similar to how the differential is obtained from the gradient. Using a function with unit norm yields the directional derivative along that function.

In physics, it's common to use the Dirac delta function in place of a generic test function, for yielding the functional derivative at the point (this is a point of the whole functional derivative as a partial derivative is a component of the gradient):

$\frac{\delta F}{\delta \varphi(y)}=\lim_{\varepsilon\to 0}\frac{F-F}{\varepsilon}.$

This works in cases when formally can be expanded as a series (or at least up to first order) in . The formula is however not mathematically rigorous, since is usually not even defined.

Read more about Functional Derivative: Formal Description, Properties, Using The Delta Function As A Test Function, Examples

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