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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 any functional F mapping (continuous/smooth/with certain boundary conditions/etc.) functions φ from a manifold M to 
or 
, then, provided the following derivative exists, the functional derivative
![\frac{\delta F}{\delta \phi}[\phi]](a/../upload/math/c15800b127008a8f48e2b6ef1ab10e4e.png)
is a distribution such that for all test functions f,
![\left(\frac{\delta F}{\delta \phi}[\phi]\right)[f]=\frac{d}{d\epsilon}F[\phi+\epsilon f].](a/../upload/math/aa27da1e0c0b6049d5d55bbeeb0c5702.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}.](a/../upload/math/86760a7085df0dd8bd482e6d99228a42.png)
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