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
\mathbb{R}
or
\mathbb{C}
, then, provided the following
derivative
exists, the
functional derivative
\frac{\delta F}{\delta \phi}
\phi
is a
distribution
such that for all
test functions
f,
\left(\frac{\delta F}{\delta
\phi}
\phi
\right)
f
=\frac{d}{d\epsilon}F
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(y)
}{\varepsilon}.
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