Editing Functional derivative

{{Short description|Concept in calculus of variation}}
In the [[calculus of variations]], a field of [[mathematical analysis]], the '''functional derivative''' (or '''variational derivative''')<ref name="GiaquintaHildebrandtP18">{{harvp|Giaquinta|Hildebrandt|1996|p=18}}</ref> relates a change in a [[Functional (mathematics)|functional]] (a functional in this sense is a function that acts on functions) to a change in a [[Function (mathematics)|function]] on which the functional depends.

In the calculus of variations, functionals are usually expressed in terms of an [[integral]] of functions, their [[Argument of a function|arguments]], and their [[derivative]]s. In an integrand {{math|''L''}} of a functional, if a function {{math|''f''}} is varied by adding to it another function {{math|''δf''}} that is arbitrarily small, and the resulting integrand is expanded in powers of {{math|''δf''}}, the coefficient of {{math|''δf''}} in the first order term is called the functional derivative.

For example, consider the functional
<math display="block"> J[f] = \int_a^b L( \, x, f(x), f'{(x)} \, ) \, dx \, , </math>
where {{math|''f'' &prime;(''x'') &equiv; ''df''/''dx''}}. If {{math|''f''}} is varied by adding to it a function {{math|''δf''}}, and the resulting integrand {{math|''L''(''x'', ''f'' +''δf'', ''f'' &prime;+''δf'' &prime;)}} is expanded in powers of {{math|''δf''}}, then the change in the value of {{math|''J''}} to first order in {{math|''δf''}} can be expressed as follows:<ref name="GiaquintaHildebrandtP18" /><ref Group = 'Note'>According to {{Harvp|Giaquinta|Hildebrandt|1996|p=18}}, this notation is customary in [[Physics|physical]] literature.</ref>
<math display="block">\begin{align}
\delta J &= \int_a^b \left( \frac{\partial L}{\partial f} \delta f(x) + \frac{\partial L}{\partial f'} \frac{d}{dx} \delta f(x) \right) \, dx \, \\[1ex]
&= \int_a^b \left( \frac{\partial L}{\partial f} - \frac{d}{dx} \frac{\partial L}{\partial f'} \right) \delta f(x) \, dx \, + \, \frac{\partial L}{\partial f'} (b) \delta f(b) \, - \, \frac{\partial L}{\partial f'} (a) \delta f(a)
\end{align} </math>
where the variation in the derivative, {{math|''δf'' &prime;}} was rewritten as the derivative of the variation {{math|(''δf'') &prime;}}, and [[integration by parts]] was used in these derivatives.

==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>.

==Properties==
Like the derivative of a function, the functional derivative satisfies the following properties, where {{math|''F''[''ρ'']}} and {{math|''G''[''ρ'']}} are functionals:<ref group="Note">
Here the notation
<math display="block">\frac{\delta{F}}{\delta\rho}(x) \equiv \frac{\delta{F}}{\delta\rho(x)}</math>
is introduced.
</ref>
* Linearity:<ref name=ParrYangP247A.3>{{harvp|Parr|Yang|1989|loc= p. 247, Eq. A.3}}.</ref> <math display="block">\frac{\delta(\lambda F + \mu G)[\rho ]}{\delta \rho(x)} = \lambda \frac{\delta F[\rho]}{\delta \rho(x)} + \mu \frac{\delta G[\rho]}{\delta \rho(x)},</math> where {{math|''λ'', ''μ''}} are constants.
* Product rule:<ref name=ParrYangP247A.4>{{harvp|Parr|Yang|1989|loc= p. 247, Eq. A.4}}.</ref> <math display="block">\frac{\delta(FG)[\rho]}{\delta \rho(x)} = \frac{\delta F[\rho]}{\delta \rho(x)} G[\rho] + F[\rho] \frac{\delta G[\rho]}{\delta \rho(x)} \, , </math>
* Chain rules:
**If {{math|''F''}} is a functional and {{math|''G''}} another functional, then<ref>{{harvp|Greiner|Reinhardt|1996|loc=p. 38, Eq. 6}}.</ref> <math display="block">\frac{\delta F[G[\rho]] }{\delta\rho(y)} = \int dx \frac{\delta F[G]}{\delta G(x)}_{G = G[\rho]}\cdot\frac {\delta G[\rho](x)} {\delta\rho(y)} \ . </math>
**If {{math|''G''}} is an ordinary differentiable function (local functional) {{math|''g''}}, then this reduces to<ref>{{harvp|Greiner|Reinhardt|1996|loc=p. 38, Eq. 7}}.</ref> <math display="block">\frac{\delta F[g(\rho)] }{\delta\rho(y)} = \frac{\delta F[g(\rho)]}{\delta g[\rho(y) ]} \ \frac {dg(\rho)} {d\rho(y)} \ . </math>

==Determining functional derivatives==
A formula to determine functional derivatives for a common class of functionals can be written as the integral of a function and its derivatives. This is a generalization of the [[Euler–Lagrange equation]]: indeed, the functional derivative was introduced in [[physics]] within the derivation of the [[Joseph-Louis Lagrange|Lagrange]] equation of the second kind from the [[principle of least action]] in [[Lagrangian mechanics]] (18th century). The first three examples below are taken from [[density functional theory]] (20th century), the fourth from [[statistical mechanics]] (19th century).

===Formula===
Given a functional
<math display="block">F[\rho] = \int f( \boldsymbol{r}, \rho(\boldsymbol{r}), \nabla\rho(\boldsymbol{r}) )\, d\boldsymbol{r},</math>
and a function <math>\phi(\boldsymbol{r})</math> that vanishes on the boundary of the region of integration, from a previous section [[#Definition|Definition]],
<math display="block">\begin{align}
\int \frac{\delta F}{\delta\rho(\boldsymbol{r})} \, \phi(\boldsymbol{r}) \, d\boldsymbol{r}
& = \left [ \frac{d}{d\varepsilon} \int f( \boldsymbol{r}, \rho + \varepsilon \phi, \nabla\rho+\varepsilon\nabla\phi )\, d\boldsymbol{r} \right ]_{\varepsilon=0} \\
& = \int \left( \frac{\partial f}{\partial\rho} \, \phi + \frac{\partial f}{\partial\nabla\rho} \cdot \nabla\phi \right) d\boldsymbol{r} \\
& = \int \left[ \frac{\partial f}{\partial\rho} \, \phi + \nabla \cdot \left( \frac{\partial f}{\partial\nabla\rho} \, \phi \right) - \left( \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right) \phi \right] d\boldsymbol{r} \\
& = \int \left[ \frac{\partial f}{\partial\rho} \, \phi - \left( \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right) \phi \right] d\boldsymbol{r} \\
& = \int \left( \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial\nabla\rho} \right) \phi(\boldsymbol{r}) \ d\boldsymbol{r} \, .
\end{align}</math>

The second line is obtained using the [[total derivative]], where {{math|''∂f'' /''∂∇ρ''}} is a [[Matrix calculus#Scalar-by-vector|derivative of a scalar with respect to a vector]].<ref group="Note">For a three-dimensional Cartesian coordinate system,
<math display="block">\frac{\partial f}{\partial\nabla\rho} = \frac{\partial f}{\partial\rho_x} \mathbf{\hat{i}} + \frac{\partial f}{\partial\rho_y} \mathbf{\hat{j}} + \frac{\partial f}{\partial\rho_z} \mathbf{\hat{k}}\, ,</math>
where <math>\rho_x = \frac{\partial \rho}{\partial x}\, , \ \rho_y = \frac{\partial \rho}{\partial y}\, , \ \rho_z = \frac{\partial \rho}{\partial z}</math> and <math>\mathbf{\hat{i}}</math>, <math>\mathbf{\hat{j}}</math>, <math>\mathbf{\hat{k}}</math> are unit vectors along the x, y, z axes.</ref>

The third line was obtained by use of a [[Divergence#Properties|product rule for divergence]]. The fourth line was obtained using the [[divergence theorem]] and the condition that <math>\phi=0</math> on the boundary of the region of integration. Since <math>\phi</math> is also an arbitrary function, applying the [[fundamental lemma of calculus of variations]] to the last line, the functional derivative is
<math display="block">\frac{\delta F}{\delta\rho(\boldsymbol{r})} = \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial\nabla\rho} </math>

where {{math|1=''ρ'' = ''ρ''('''''r''''')}} and {{math|1=''f'' = ''f'' ('''''r''''', ''ρ'', &nabla;''ρ'')}}. This formula is for the case of the functional form given by {{math|''F''[''ρ'']}} at the beginning of this section. For other functional forms, the definition of the functional derivative can be used as the starting point for its determination. (See the example [[#Coulomb potential energy functional|Coulomb potential energy functional]].)

The above equation for the functional derivative can be generalized to the case that includes higher dimensions and higher order derivatives. The functional would be,
<math display="block">F[\rho(\boldsymbol{r})] = \int f( \boldsymbol{r}, \rho(\boldsymbol{r}), \nabla\rho(\boldsymbol{r}), \nabla^{(2)}\rho(\boldsymbol{r}), \dots, \nabla^{(N)}\rho(\boldsymbol{r}))\, d\boldsymbol{r},</math>

where the vector {{math|'''''r''''' &isin; '''R'''<sup>''n''</sup>}}, and {{math|&nabla;<sup>(''i'')</sup>}} is a tensor whose {{math|''n<sup>i</sup>''}} components are partial derivative operators of order {{math|''i''}},
<math display="block"> \left [ \nabla^{(i)} \right ]_{\alpha_1 \alpha_2 \cdots \alpha_i} = \frac {\partial^{\, i}} {\partial r_{\alpha_1} \partial r_{\alpha_2} \cdots \partial r_{\alpha_i} } \qquad \qquad \text{where} \quad \alpha_1, \alpha_2, \dots, \alpha_i = 1, 2, \dots , n \ . </math><ref group="Note">For example, for the case of three dimensions ({{math|1=''n'' = 3}}) and second order derivatives ({{math|1=''i'' = 2}}), the tensor {{math|&nabla;<sup>(2)</sup>}} has components,
<math display="block"> \left [ \nabla^{(2)} \right ]_{\alpha \beta} = \frac {\partial^{\,2}} {\partial r_{\alpha} \, \partial r_{\beta}} </math>where <math>\alpha</math> and <math>\beta</math> can be <math>1,2,3</math>.</ref>

An analogous application of the definition of the functional derivative yields
<math display="block">\begin{align}
\frac{\delta F[\rho]}{\delta \rho} &{} = \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial(\nabla\rho)} + \nabla^{(2)} \cdot \frac{\partial f}{\partial\left(\nabla^{(2)}\rho\right)} + \dots + (-1)^N \nabla^{(N)} \cdot \frac{\partial f}{\partial\left(\nabla^{(N)}\rho\right)} \\
&{} = \frac{\partial f}{\partial\rho} + \sum_{i=1}^N (-1)^{i}\nabla^{(i)} \cdot \frac{\partial f}{\partial\left(\nabla^{(i)}\rho\right)} \ .
\end{align}</math>

In the last two equations, the {{math|''n<sup>i</sup>''}} components of the tensor <math> \frac{\partial f}{\partial\left(\nabla^{(i)}\rho\right)} </math> are partial derivatives of {{math|''f''}} with respect to partial derivatives of ''ρ'',
<math display="block"> \left [ \frac {\partial f} {\partial \left (\nabla^{(i)}\rho \right ) } \right ]_{\alpha_1 \alpha_2 \cdots \alpha_i} = \frac {\partial f} {\partial \rho_{\alpha_1 \alpha_2 \cdots \alpha_i} }  </math>
where <math>  \rho_{\alpha_1 \alpha_2 \cdots \alpha_i} \equiv \frac {\partial^{\,i}\rho} {\partial r_{\alpha_1} \, \partial r_{\alpha_2} \cdots \partial r_{\alpha_i} } </math>, and the tensor scalar product is,
<math display="block"> \nabla^{(i)} \cdot \frac{\partial f}{\partial\left(\nabla^{(i)}\rho\right)} = \sum_{\alpha_1, \alpha_2, \cdots, \alpha_i = 1}^n \ \frac {\partial^{\, i} } {\partial r_{\alpha_1} \, \partial r_{\alpha_2} \cdots \partial r_{\alpha_i} } \ \frac {\partial f} {\partial \rho_{\alpha_1 \alpha_2 \cdots \alpha_i} } \ . </math> <ref group="Note">For example, for the case {{math|1=''n'' = 3}} and {{math|1=''i'' = 2}}, the tensor scalar product is,
<math display="block"> \nabla^{(2)} \cdot \frac{\partial f}{\partial\left(\nabla^{(2)}\rho\right)} = \sum_{\alpha, \beta = 1}^3 \ \frac {\partial^{\, 2} } {\partial r_{\alpha} \, \partial r_{\beta} } \, \frac {\partial f} {\partial \rho_{\alpha \beta} } , </math>where <math>\rho_{\alpha \beta} \equiv \frac {\partial^{\, 2}\rho} {\partial r_{\alpha} \, \partial r_{\beta} }</math>.</ref>

===Examples===

====Thomas–Fermi kinetic energy functional====
The [[Thomas–Fermi model]] of 1927 used a kinetic energy functional for a noninteracting uniform [[free electron model|electron gas]] in a first attempt of [[density-functional theory]] of electronic structure:
<math display="block">T_\mathrm{TF}[\rho] = C_\mathrm{F} \int \rho^{5/3}(\mathbf{r}) \, d\mathbf{r} \, .</math>
Since the integrand of {{math|''T''<sub>TF</sub>[''ρ'']}} does not involve derivatives of {{math|''ρ''('''''r''''')}}, the functional derivative of {{math|''T''<sub>TF</sub>[''ρ'']}} is,<ref name=ParrYangP247A.6>{{harvp|Parr|Yang|1989|loc=p. 247, Eq. A.6}}.</ref>
<math display="block">\frac{\delta T_{\mathrm{TF}}}{\delta \rho (\boldsymbol{r}) }
= C_\mathrm{F} \frac{\partial \rho^{5/3}(\mathbf{r})}{\partial \rho(\mathbf{r})}
= \frac{5}{3} C_\mathrm{F} \rho^{2/3}(\mathbf{r}) \, .</math>

====Coulomb potential energy functional====
The '''electron-nucleus''' potential energy is
<math display="block">V[\rho] = \int \frac{\rho(\boldsymbol{r})}{|\boldsymbol{r}|} \ d\boldsymbol{r}.</math>

Applying the definition of functional derivative,
<math display="block">\begin{align}
\int \frac{\delta V}{\delta \rho(\boldsymbol{r})} \ \phi(\boldsymbol{r}) \ d\boldsymbol{r}
& {} = \left [ \frac{d}{d\varepsilon} \int \frac{\rho(\boldsymbol{r}) + \varepsilon \phi(\boldsymbol{r})}{|\boldsymbol{r}|} \ d\boldsymbol{r} \right ]_{\varepsilon=0} \\[1ex]
& {} = \int \frac {\phi(\boldsymbol{r})} {|\boldsymbol{r}|} \ d\boldsymbol{r} \, .
\end{align}</math>
So,
<math display="block"> \frac{\delta V}{\delta \rho(\boldsymbol{r})} = \frac{1}{|\boldsymbol{r}|} \ . </math>

The functional derivative of the classical part of the '''electron-electron interaction''' (often called Hartree energy) is 
<math display="block">J[\rho] = \frac{1}{2}\iint \frac{\rho(\mathbf{r}) \rho(\mathbf{r}')}{| \mathbf{r}-\mathbf{r}' |}\, d\mathbf{r} d\mathbf{r}' \, .</math>
From the [[#Functional derivative|definition of the functional derivative]],
<math display="block">\begin{align}
\int \frac{\delta J}{\delta\rho(\boldsymbol{r})} \phi(\boldsymbol{r})d\boldsymbol{r}
& {} = \left [ \frac {d \ }{d\varepsilon} \, J[\rho + \varepsilon\phi] \right ]_{\varepsilon = 0} \\
& {} = \left [ \frac {d \ }{d\varepsilon} \, \left ( \frac{1}{2}\iint \frac {[\rho(\boldsymbol{r}) + \varepsilon \phi(\boldsymbol{r})] \, [\rho(\boldsymbol{r}') + \varepsilon \phi(\boldsymbol{r}')] }{| \boldsymbol{r}-\boldsymbol{r}' |}\, d\boldsymbol{r} d\boldsymbol{r}' \right ) \right ]_{\varepsilon = 0} \\
& {} = \frac{1}{2}\iint \frac {\rho(\boldsymbol{r}') \phi(\boldsymbol{r}) }{| \boldsymbol{r}-\boldsymbol{r}' |}\, d\boldsymbol{r} d\boldsymbol{r}' + \frac{1}{2}\iint \frac {\rho(\boldsymbol{r}) \phi(\boldsymbol{r}') }{| \boldsymbol{r}-\boldsymbol{r}' |}\, d\boldsymbol{r} d\boldsymbol{r}' \\
\end{align}</math>
The first and second terms on the right hand side of the last equation are equal, since {{math|'''''r'''''}} and {{math|'''''r&prime;'''''}} in the second term can be interchanged without changing the value of the integral. Therefore,
<math display="block"> \int \frac{\delta J}{\delta\rho(\boldsymbol{r})} \phi(\boldsymbol{r})d\boldsymbol{r} = \int \left ( \int \frac {\rho(\boldsymbol{r}') }{| \boldsymbol{r}-\boldsymbol{r}' |} d\boldsymbol{r}' \right ) \phi(\boldsymbol{r}) d\boldsymbol{r} </math>
and the functional derivative of the electron-electron Coulomb potential energy functional {{math|''J''}}[''ρ''] is,<ref name=ParrYangP248A.11>{{harvp|Parr|Yang|1989|loc=p. 248, Eq. A.11}}.</ref>
<math display="block"> \frac{\delta J}{\delta\rho(\boldsymbol{r})} = \int \frac {\rho(\boldsymbol{r}') }{| \boldsymbol{r}-\boldsymbol{r}' |} d\boldsymbol{r}' \, . </math>

The second functional derivative is
<math display="block">\frac{\delta^2 J[\rho]}{\delta \rho(\mathbf{r}')\delta\rho(\mathbf{r})} = \frac{\partial}{\partial \rho(\mathbf{r}')} \left ( \frac{\rho(\mathbf{r}')}{| \mathbf{r}-\mathbf{r}' |} \right ) = \frac{1}{| \mathbf{r}-\mathbf{r}' |}.</math>

====von Weizsäcker kinetic energy functional====
In 1935 [[Carl Friedrich von Weizsacker|von Weizsäcker]] proposed to add a gradient correction to the Thomas-Fermi kinetic energy functional to make it better suit a molecular electron cloud:
<math display="block">T_\mathrm{W}[\rho] = \frac{1}{8} \int \frac{\nabla\rho(\mathbf{r}) \cdot \nabla\rho(\mathbf{r})}{ \rho(\mathbf{r}) } d\mathbf{r} = \int t_\mathrm{W}(\mathbf{r}) \ d\mathbf{r} \, ,</math>
where
<math display="block"> t_\mathrm{W} \equiv \frac{1}{8} \frac{\nabla\rho \cdot \nabla\rho}{ \rho } \qquad \text{and} \ \ \rho = \rho(\boldsymbol{r}) \ . </math>
Using a previously derived [[#Formula|formula]] for the functional derivative,
<math display="block">\begin{align}
\frac{\delta T_\mathrm{W}}{\delta \rho}
& = \frac{\partial t_\mathrm{W}}{\partial \rho} - \nabla\cdot\frac{\partial t_\mathrm{W}}{\partial \nabla \rho} \\
& = -\frac{1}{8}\frac{\nabla\rho \cdot \nabla\rho}{\rho^2} - \left ( \frac {1}{4} \frac {\nabla^2\rho} {\rho} - \frac {1}{4} \frac {\nabla\rho \cdot \nabla\rho} {\rho^2} \right ) \qquad \text{where} \ \ \nabla^2 = \nabla \cdot \nabla \ ,
\end{align}</math>
and the result is,<ref name=ParrYangP247A.9>{{harvp|Parr|Yang|1989|loc= p. 247, Eq. A.9}}.</ref>
<math display="block"> \frac{\delta T_\mathrm{W}}{\delta \rho} = \ \ \, \frac{1}{8}\frac{\nabla\rho \cdot \nabla\rho}{\rho^2} - \frac{1}{4}\frac{\nabla^2\rho}{\rho} \ . </math>

====Entropy====
The [[information entropy|entropy]] of a discrete [[random variable]] is a functional of the [[probability mass function]].

<math display="block">H[p(x)] = -\sum_x p(x) \log p(x)</math>
Thus,
<math display="block">\begin{align}
\sum_x \frac{\delta H}{\delta p(x)} \, \phi(x)
& {} = \left[ \frac{d}{d\varepsilon} H[p(x) + \varepsilon\phi(x)] \right]_{\varepsilon=0}\\
& {} = \left [- \, \frac{d}{d\varepsilon} \sum_x \, [p(x) + \varepsilon\phi(x)] \ \log [p(x) + \varepsilon\phi(x)] \right]_{\varepsilon=0} \\
& {} = -\sum_x \, [1+\log p(x)] \ \phi(x) \, .
\end{align}</math>
Thus,
<math display="block">\frac{\delta H}{\delta p(x)} = -1-\log p(x).</math>

==== Exponential ====

Let
<math display="block"> F[\varphi(x)]= e^{\int \varphi(x) g(x)dx}.</math>

Using the delta function as a test function,
<math display="block">\begin{align}
\frac{\delta F[\varphi(x)]}{\delta \varphi(y)}
& {} = \lim_{\varepsilon\to 0}\frac{F[\varphi(x)+\varepsilon\delta(x-y)]-F[\varphi(x)]}{\varepsilon}\\
& {} = \lim_{\varepsilon\to 0}\frac{e^{\int (\varphi(x)+\varepsilon\delta(x-y)) g(x)dx}-e^{\int \varphi(x) g(x)dx}}{\varepsilon}\\
& {} = e^{\int \varphi(x) g(x)dx}\lim_{\varepsilon\to 0}\frac{e^{\varepsilon \int \delta(x-y) g(x)dx}-1}{\varepsilon}\\
& {} = e^{\int \varphi(x) g(x)dx}\lim_{\varepsilon\to 0}\frac{e^{\varepsilon g(y)}-1}{\varepsilon}\\
& {} = e^{\int \varphi(x) g(x)dx}g(y).
\end{align}</math>

Thus,
<math display="block"> \frac{\delta F[\varphi(x)]}{\delta \varphi(y)} = g(y) F[\varphi(x)]. </math>

This is particularly useful in calculating the [[Correlation function (quantum field theory)|correlation functions]] from the [[Partition function (quantum field theory)|partition function]] in [[quantum field theory]].

====Functional derivative of a function====
A function can be written in the form of an integral like a functional. For example,
<math display="block">\rho(\boldsymbol{r}) = F[\rho] = \int \rho(\boldsymbol{r}') \delta(\boldsymbol{r}-\boldsymbol{r}')\, d\boldsymbol{r}'.</math>
Since the integrand does not depend on derivatives of ''ρ'', the functional derivative of ''ρ''{{math|('''''r''''')}} is,
<math display="block">\frac {\delta \rho(\boldsymbol{r})} {\delta\rho(\boldsymbol{r}')} \equiv \frac {\delta F} {\delta\rho(\boldsymbol{r}')}
= \frac{\partial \ \ }{\partial \rho(\boldsymbol{r}')} \, [\rho(\boldsymbol{r}') \delta(\boldsymbol{r}-\boldsymbol{r}')]
= \delta(\boldsymbol{r}-\boldsymbol{r}').</math>

==== Functional derivative of iterated function====
The functional derivative of the iterated function <math>f(f(x))</math> is given by:
<math display="block">\frac{\delta f(f(x))}{\delta f(y) } = f'(f(x))\delta(x-y) + \delta(f(x)-y)</math>
and
<math display="block">\frac{\delta f(f(f(x)))}{\delta f(y) } = f'(f(f(x))(f'(f(x))\delta(x-y) + \delta(f(x)-y)) + \delta(f(f(x))-y)</math>

In general:
<math display="block">\frac{\delta f^N(x)}{\delta f(y)} = f'( f^{N-1}(x) ) \frac{ \delta f^{N-1}(x)}{\delta f(y)} + \delta( f^{N-1}(x) - y ) </math>

Putting in {{math|1=''N'' = 0}} gives:
<math display="block"> \frac{\delta f^{-1}(x)}{\delta f(y) } = - \frac{ \delta(f^{-1}(x)-y ) }{ f'(f^{-1}(x)) }</math>

==Using the delta function as a test function==
In physics, it is common to use the [[Dirac delta function]] <math>\delta(x-y)</math> in place of a generic test function <math>\phi(x)</math>, for yielding the functional derivative at the point <math>y</math> (this is a point of the whole functional derivative as a [[partial derivative]] is a component of the gradient):<ref>{{harvp|Greiner|Reinhardt|1996|p=37}}</ref>
<math display="block">\frac{\delta F[\rho(x)]}{\delta \rho(y)}=\lim_{\varepsilon\to 0}\frac{F[\rho(x)+\varepsilon\delta(x-y)]-F[\rho(x)]}{\varepsilon}.</math>

This works in cases when <math>F[\rho(x)+\varepsilon f(x)]</math> formally can be expanded as a series (or at least up to first order) in <math>\varepsilon</math>. The formula is however not mathematically rigorous, since <math>F[\rho(x)+\varepsilon\delta(x-y)]</math> is usually not even defined.

The definition given in a previous section is based on a relationship that holds for all test functions <math>\phi(x)</math>, so one might think that it should hold also when <math>\phi(x)</math> is chosen to be a specific function such as the [[Dirac delta function|delta function]]. However, the latter is not a valid test function (it is not even a proper function).

In the definition, the functional derivative describes how the functional <math>F[\rho(x)]</math> changes as a result of a small change in the entire function <math>\rho(x)</math>. The particular form of the change in <math>\rho(x)</math> is not specified, but it should stretch over the whole interval on which <math>x</math> is defined. Employing the particular form of the perturbation given by the delta function has the meaning that <math>\rho(x)</math> is varied only in the point <math>y</math>. Except for this point, there is no variation in <math>\rho(x)</math>.

==Notes==
{{Reflist|group=Note}}

==Footnotes==
{{reflist|29em}}

==References==
*{{cite book | last1=Courant | first1=Richard | author-link1=Richard Courant | last2=Hilbert | first2=David | author-link2=David Hilbert | title = Methods of Mathematical Physics | volume = I | edition = First English | publisher = [[Interscience Publishers]], Inc | year = 1953 | location = New York, New York | chapter = Chapter IV. The Calculus of Variations | pages = 164–274 | isbn = 978-0471504474| mr = 0065391 | zbl = 0001.00501}}.
*{{Citation
 | last1 = Frigyik
 | first1 = Béla A.
 | last2 = Srivastava
 | first2 = Santosh
 | last3 = Gupta
 | first3 = Maya R.
 | title = Introduction to Functional Derivatives
 | place = Seattle, WA
 | publisher = Department of Electrical Engineering at the University of Washington
 | series = UWEE Tech Report
 | volume = UWEETR-2008-0001
 | date = January 2008
 | pages = 7
 | url = https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2008-0001.pdf
 | access-date = 2013-10-23
 | archive-url = https://web.archive.org/web/20170217025324/https://www2.ee.washington.edu/techsite/papers/documents/UWEETR-2008-0001.pdf
 | archive-date = 2017-02-17
 | url-status = dead
 }}.
*{{Citation
 | last1 = Gelfand
 | first1 = I. M.
 | author-link = Israel Gelfand
 | last2 = Fomin
 | first2 = S. V.
 | author2-link = Sergei Fomin
 | title = Calculus of variations
 | place = Mineola, N.Y.
 | publisher = [[Dover Publications]]
 | series = translated and edited by Richard A. Silverman
 | orig-year = 1963
 | year = 2000
 | edition = Revised English
 | url = http://store.doverpublications.com/0486414485.html
 | isbn = 978-0486414485
 | mr = 0160139
 | zbl = 0127.05402
}}.
*{{Citation
 | last1 = Giaquinta
 | first1 = Mariano
 | author-link = Mariano Giaquinta
 | last2 = Hildebrandt
 | first2 = Stefan
 | title = Calculus of Variations 1. The Lagrangian Formalism
 | place = Berlin
 | publisher = [[Springer-Verlag]]
 | series = Grundlehren der Mathematischen Wissenschaften
 | volume = 310
 | year = 1996
 | edition = 1st
 | isbn = 3-540-50625-X
 | mr = 1368401
 | zbl = 0853.49001
}}.
*{{Citation
 | last1 = Greiner
 | first1 = Walter
 | author-link1 = Walter Greiner
 | last2 = Reinhardt
 | first2 = Joachim
 | title = Field quantization
 | place = Berlin–Heidelberg–New York
 | publisher = Springer-Verlag
 | series = With a foreword by D. A. Bromley
 | year = 1996
 | chapter = Section 2.3 – Functional derivatives
 | pages = [https://archive.org/details/fieldquantizatio0000grei/page/36 36–38]
 | chapter-url = https://archive.org/details/fieldquantizatio0000grei/page/36
 | isbn = 3-540-59179-6
 | mr = 1383589
 | zbl = 0844.00006
 }}.
*{{cite book |first1=R. G.|last1=Parr|first2=W.|last2=Yang| title = Density-Functional Theory of Atoms and Molecules | chapter = Appendix A, Functionals | pages = 246–254 | publisher = Oxford University Press | year = 1989 |location=New York| url = https://books.google.com/books?id=mGOpScSIwU4C&q=Density-Functional+Theory+of+Atoms+and+Molecules | isbn = 978-0195042795}}

==External links==
* {{springer|title=Functional derivative|id=p/f042040}}

{{Functional analysis}}
{{Analysis in topological vector spaces}}

[[Category:Calculus of variations]]
[[Category:Differential calculus]]
[[Category:Differential operators]]
[[Category:Topological vector spaces]]
[[Category:Variational analysis]]