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