Editing Functional derivative (section)

==Definition==

In this section, the functional differential (or variation or first variation)<Ref Group = 'Note'> Called ''first variation'' in {{harv|Giaquinta|Hildebrandt|1996|p=3}}, ''variation'' or ''first variation'' in {{harv|Courant|Hilbert|1953|p=186}}, ''variation'' or ''differential'' in {{harv|Gelfand|Fomin|2000|loc= p. 11, § 3.2}} and ''differential'' in {{harv|Parr|Yang|1989|p=246}}.</ref> is defined. Then the functional derivative is defined in terms of the functional differential.

===Functional differential===

Suppose <math>B</math> is a [[Banach space]] and <math>F</math> is a [[Functional (mathematics)|functional]] defined on <math>B</math>.
The differential of <math>F</math> at a point <math>\rho\in B</math> is the [[linear functional]] <math>\delta F[\rho,\cdot]</math> on <math>B</math> defined<ref name="GelfandFominp11">{{harvp|Gelfand|Fomin|2000|p=11}}.</ref> by the condition that, for all <math>\phi\in B</math>,
<math display="block">
F[\rho+\phi] - F[\rho]
=
\delta F [\rho; \phi] + \varepsilon \left\|\phi\right\|
</math>
where <math>\varepsilon</math> is a real number that depends on <math>\|\phi\|</math> in such a way that <math>\varepsilon\to 0</math> as <math>\|\phi\|\to 0</math>. This means that <math>\delta F[\rho,\cdot]</math> is the [[Fréchet derivative]] of <math>F</math> at <math>\rho</math>.

However, this notion of functional differential is so strong it may not exist,<ref name="GiaquintaHildebrandtP180">{{harvp|Giaquinta|Hildebrandt|1996|p=10}}.</ref> and in those cases a weaker notion, like the [[Gateaux derivative]] is preferred. In many practical cases, the functional differential is defined<ref name="GiaquintaHildebrandtP3">{{harvp|Giaquinta|Hildebrandt|1996|p=10}}.</ref> as the directional derivative
<math display="block">
\begin{align}
\delta F[\rho,\phi]
&= \lim_{\varepsilon\to 0}\frac{F[\rho+\varepsilon \phi]-F[\rho]}{\varepsilon} \\[1ex]
&= \left [ \frac{d}{d\varepsilon}F[\rho+\varepsilon \phi]\right ]_{\varepsilon=0}.
\end{align}
</math>
Note that this notion of the functional differential can even be defined without a norm.

===Functional derivative===

In many applications, the domain of the functional <math>F</math> is a space of differentiable functions <math>\rho</math> defined on some space <math>\Omega</math> and <math>F</math> is of the form
<math display="block">
F[\rho]
=
\int_\Omega L(x,\rho(x),D\rho(x))\,dx
</math>
for some function <math>L(x,\rho(x),D\rho(x))</math> that may depend on <math>x</math>, the value <math>\rho(x)</math> and the derivative <math>D\rho(x)</math>.
If this is the case and, moreover, <math>\delta F[\rho,\phi]</math> can be written as the integral of <math>\phi</math> times another function (denoted {{math|''δF''/''δρ''}})
<math display="block">\delta F [\rho, \phi] = \int_\Omega \frac {\delta F} {\delta \rho}(x) \ \phi(x) \ dx</math>
then this function {{math|''δF''/''δρ''}} is called the '''functional derivative''' of {{math|''F''}} at {{math|''ρ''}}.<ref name=ParrYangP246A.2>{{harvp|Parr|Yang|1989|loc= p. 246, Eq. A.2}}.</ref><ref name=GreinerReinhardtP36.2>{{harvp|Greiner|Reinhardt|1996|p=36,37}}.</ref> If <math>F</math> is restricted to only certain functions <math>\rho</math> (for example, if there are some boundary conditions imposed) then <math>\phi</math> is restricted to  functions such that <math>\rho+\varepsilon\phi</math> continues to satisfy these conditions.

Heuristically, <math>\phi</math> is the change in <math>\rho</math>, so we 'formally' have <math>\phi = \delta\rho</math>, and then this is similar in form to the [[total differential]] of a function <math>F(\rho_1,\rho_2,\dots,\rho_n)</math>,
<math display="block"> dF = \sum_{i=1} ^n \frac {\partial F} {\partial \rho_i} \ d\rho_i ,</math>
where <math>\rho_1,\rho_2,\dots,\rho_n</math> are independent variables.
Comparing the last two equations, the functional derivative <math>\delta F/\delta\rho(x)</math> has a role similar to that of the partial derivative <math>\partial F/\partial\rho_i</math>, where the variable of integration <math>x</math> is like a continuous version of the summation index <math>i</math>.<ref name=ParrYangP246>{{harvp|Parr|Yang|1989|p=246}}.</ref> One thinks of {{math|''δF''/''δρ''}} as the gradient of {{math|''F''}} at the point {{math|''ρ''}}, so the value {{math|''δF''/''δρ(x)''}} measures how much the functional {{math|''F''}} will change if the function {{math|''ρ''}} is changed at the point {{math|''x''}}. Hence the formula
<math display="block">\int \frac{\delta F}{\delta\rho}(x) \phi(x) \; dx</math>
is regarded as the directional derivative at point <math>\rho</math> in the direction of <math>\phi</math>. This is analogous to vector calculus, where the inner product of a vector <math>v</math> with the gradient gives the directional derivative in the direction of <math>v</math>.