Editing Functional derivative (section)

===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.