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In the [[calculus of variations]], a field of [[mathematical analysis]], the '''functional derivative''' (or '''variational derivative''')<ref name="GiaquintaHildebrandtP18">{{harv|Giaquinta|Hildebrandt|1996|p=18}}</ref> relates a change in a [[Functional (mathematics)|functional]] to a change in a [[Function (mathematics)|function]] that the functional depends on.
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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 {{math|''L''}} is expanded in powers of {{math|''δf''}}, the coefficient of {{math|''δf''}} in the first order term is called the functional derivative.   
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For example, consider the functional
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:<math> J[f] = \int \limits_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 '+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 {{harvtxt|Giaquinta|Hildebrandt|1996|p=18}}, this notation is customary in [[Physics|physical]] literature.</ref>
:<math>  \delta J = \int_a^b  \frac{\delta J}{\delta f(x)} {\delta f(x)} dx \, . </math>
The coefficient of {{math|''δf(x)''}}, denoted as {{math|''δJ''/''δf(x)''}}, is called the '''functional derivative''' of {{math|''J''}} with respect to {{math|''f''}} at the point {{math|''x''}}.<ref name=ParrYangP246>{{harv|Parr|Yang|1989|p=246}}.</ref> For this example functional, the functional derivative is the left hand side of the [[Calculus_of_variations#Euler.E2.80.93Lagrange_equation|Euler-Lagrange equation]],<ref name=GelfandFominP28>{{harv|Gelfand|Fomin|2000|p=28}}</ref>
:<math> \frac{\delta J}{\delta f(x)} = \frac{\part L}{\part f} -\frac{d}{dx} \frac{\part L}{\part f'} \, . </math>


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The functional derivative is defined. Then the functional differential is defined in terms of the functional derivative.  
 
===Functional derivative===
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Given a [[manifold]] ''M'' representing ([[continuous function (topology)|continuous]]/[[smooth function|smooth]]/with certain [[boundary condition]]s/etc.) functions ''ρ'' and a [[functional (mathematics)|functional]] ''F'' defined as
 
::<math>F\colon M \rightarrow \mathbb{R} \quad \mbox{or} \quad F\colon M \rightarrow \mathbb{C} \, ,</math>  
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the '''functional derivative''' of {{math|''F''[}}''ρ''], denoted {{math|''δF/δ''}}''ρ'', is  defined by<ref name=ParrYangP246A.2>{{harv|Parr|Yang|1989|loc= p. 246, Eq. A.2}}.</ref>
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:<math>
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\begin{align}
 
\int \frac{\delta F}{\delta\rho(x)} \ \phi(x) \ dx
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&= \lim_{\varepsilon\to 0}\frac{F[\rho+\varepsilon \phi]-F[\rho]}{\varepsilon} \\
 
&= \left [ \frac{d}{d\epsilon}F[\rho+\epsilon \phi]\right ]_{\epsilon=0},
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\end{align}
 
</math>
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</ul>


where  <math>\epsilon \phi</math> is the variation of ''ρ'', and  <math>\phi</math> is an arbitrary function.
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===Functional differential===
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The differential (or variation or first variation) of the functional {{math|''F''}}[''ρ''] is,<ref name=ParrYangP246A.1> {{harv|Parr|Yang|1989|loc= p. 246, Eq. A.1}}.</ref> <Ref Group = 'Note' > Called ''differential'' in {{harv|Parr|Yang|1989|p=246}}, ''variation'' or ''first variation'' in {{harv|Courant|Hilbert|1953|p=186}}, and ''variation'' or ''differential'' in {{harv|Gelfand|Fomin|2000|loc= p. 11, &sect; 3.2}}.</ref>
 
:<math> \delta F = \int  \frac {\delta F} {\delta \rho(x)} \ \delta \rho(x)  \ dx \ , </math>
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where {{math|''δ''}}''ρ''{{math|(''x'') {{=}} ''εϕ''(''x'')}} is the variation of ''ρ''{{math|(''x'')}}. This is similar in form to the [[total differential]] of a function {{math|''F''}}(''ρ''<sub>1</sub>, ''ρ''<sub>2</sub>, ... , ''ρ''<sub>n</sub>),
 
:<math> dF =   \sum_{i=1} ^n  \frac {\partial F} {\partial \rho_i} \ d\rho_i  \ ,</math>
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where ''ρ<sub>1</sub>, ρ<sub>2</sub>, ... , ρ<sub>n</sub>'' are independent variables.  
 
Comparing the last two equations, the functional derivative {{math|''δF''/''δ''}}''ρ''{{math|(''x'')}} has a role similar to that of the partial derivative {{math|''&part;F/&part;''}}''ρ<sub>i</sub>''&nbsp;, where the variable of integration {{math|''x''}} is like a continuous version of the summation index {{math|''i''}}.<ref name=ParrYangP246 />
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</ul>


===Formal description===
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The definition of a functional derivative may be made more mathematically precise and formal by defining the [[topological vector space|space of functions]] more carefully. For example, when the space of functions is a [[Banach space]], the functional derivative becomes known as the [[Fréchet derivative]], while one uses the [[Gâteaux derivative]] on more general [[locally convex space]]s. Note that the well-known [[Hilbert space]]s are special cases of [[Banach space]]s. The more formal treatment allows many theorems from ordinary [[calculus]] and [[Mathematical analysis|analysis]] to be generalized to corresponding theorems in [[functional analysis]], as well as numerous new theorems to be stated.


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Like the derivative of a function, the functional derivative satisfies the following properties, where {{math|''F''}}[''ρ''] and {{math|''G''}}[''ρ''] are functionals:
 
* Linear:<ref name=ParrYangP247A.3>{{harv|Parr|Yang|1989|loc= p. 247, Eq. A.3}}.</ref>
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: <math>\frac{\delta(\lambda F + \mu G)}{\delta \rho(x)} = \lambda \frac{\delta F}{\delta \rho(x)} + \mu \frac{\delta G}{\delta \rho(x)},\ \qquad  \lambda,\mu</math> &nbsp; [[Constant (mathematics)|constant]],
 
* Product rule:<ref name=ParrYangP247A.4>{{harv|Parr|Yang|1989|loc= p. 247, Eq. A.4}}.</ref>
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: <math>\frac{\delta(FG)}{\delta \rho(x)} = \frac{\delta F}{\delta \rho(x)} G + F \frac{\delta G}{\delta \rho(x)} \, , </math>
 
* Chain rules:
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:If {{math|''f''}} is a differentiable function, then
 
:<math>\displaystyle\frac{\delta F[f(\rho)] }{\delta\rho(x)}  = \frac{\delta F[f(\rho)]}{\delta f(\rho(x) )} \ \frac {df(\rho(x))} {d\rho(x)} \ , </math><ref name='ChainRule'> {{harv|Greiner|Reinhardt|1996|loc=p. 38, Eq. 7}}.</ref>  
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: <math>\frac{\delta f( F[\rho])}{\delta\rho(x)} = \frac {df(F[\rho])} {dF[\rho]} \ \frac{\delta F[\rho]}{\delta\rho(x)} \, . </math><ref name=ParrYangP251A.34>{{harv|Parr|Yang|1989|loc=p. 251, Eq. A.34}}.</ref>
 
</ul>


==Determining functional derivatives==
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We give a formula to determine functional derivatives for a common class of functionals that 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===
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Given a functional
 
:<math>F[\rho] = \int f( \boldsymbol{r}, \rho(\boldsymbol{r}), \nabla\rho(\boldsymbol{r}) )\, d\boldsymbol{r},</math>
  <li>[http://www.qzjmych.com/news/html/?84332.html http://www.qzjmych.com/news/html/?84332.html]</li>
and a function {{math|''ϕ''}}('''{{math|''r''}}''') that vanishes on the boundary of the region of integration, from a previous section [[Functional_derivative#Definition|Definition]],  
 
  <li>[http://www.3gugu.com/bbs/forum.php?mod=viewthread&tid=2065961 http://www.3gugu.com/bbs/forum.php?mod=viewthread&tid=2065961]</li>
 
  <li>[http://enseignement-lsf.com/spip.php?article64#forum18184226 http://enseignement-lsf.com/spip.php?article64#forum18184226]</li>
 
  <li>[http://assetaaa.com/en/forum.php?mod=viewthread&tid=217846 http://assetaaa.com/en/forum.php?mod=viewthread&tid=217846]</li>
 
  </ul>


:<math>
== Basically Ray Ban Aviators Small ==
\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|''&part;f'' /''&part;&nabla;''}}''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,
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:<math>
 
\begin{align}
  <li>[http://www.youxisky.com/bbs/forum.php?mod=viewthread&tid=776291 http://www.youxisky.com/bbs/forum.php?mod=viewthread&tid=776291]</li>
\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}}\, ,  \qquad
 
& \text{where} \ \rho_x = \frac{\partial \rho}{\partial x}\, ,  \ \rho_y = \frac{\partial \rho}{\partial y}\, , \ \rho_z = \frac{\partial \rho}{\partial z}\, \\
  <li>[http://www.wowhealer.com/forums/viewtopic.php?pid=932157#p932157 http://www.wowhealer.com/forums/viewtopic.php?pid=932157#p932157]</li>
& \text{and} \ \ \mathbf{\hat{i}}, \ \mathbf{\hat{j}}, \ \mathbf{\hat{k}} \ \ \text {are unit vectors along the x, y, z axes.}
 
\end{align}
  <li>[http://www.xuhui365.com/thread-1060000-1-1.html http://www.xuhui365.com/thread-1060000-1-1.html]</li>
</math>  </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|''ϕ''{{=}}0}} on the boundary of the region of integration. Since {{math|''ϕ''}} is also an arbitrary function, applying the [[fundamental lemma of calculus of variations]] to the last line, the functional derivative is
    
 
  <li>[http://www.intelliplanfpi.com/index.php?option=com_kunena&func=view&catid=7&id=47049&Itemid=53#47049 http://www.intelliplanfpi.com/index.php?option=com_kunena&func=view&catid=7&id=47049&Itemid=53#47049]</li>
:<div style="{{divstylewhite}}; width:20em; margin:.3em"><center><math>
 
\frac{\delta F}{\delta\rho(\boldsymbol{r})} = \frac{\partial f}{\partial\rho} - \nabla \cdot \frac{\partial f}{\partial\nabla\rho}
</ul>
</math></center></div>
 
where ''ρ'' = ''ρ''('''{{math|''r''}}''') and {{math|''f'' {{=}} ''f'' ('''{{math|''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 [[Functional_derivative#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>
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> \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, \cdots, \alpha_i = 1, 2, \cdots , n \ . </math><ref group = 'Note'>For example, for the case of three dimensions ({{math|''n'' {{=}} 3}}) and second order derivatives ({{math|''i'' {{=}} 2}}), the tensor {{math|&nabla;<sup>(2)</sup>}} has components,
:<math> \left [ \nabla^{(2)} \right ]_{\alpha \beta} = \frac {\partial^{\,2}} {\partial r_{\alpha} \, \partial r_{\beta}} \qquad \qquad \text{where} \quad \alpha, \beta = 1, 2, 3 \, . </math> </ref>
 
An analogous application of the definition of the functional derivative yields
 
:<math>
\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> \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} } \qquad \qquad \text{where} \quad \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> \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|''n'' {{=}} 3}} and {{math|''i'' {{=}} 2}}, the tensor scalar product is,
:<math> \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} }    \qquad \text{where} \ \ \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>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 ''ρ''{{math|('''''r''''')}}, the functional derivative of {{math|''T''<sub>TF</sub>}}[''ρ''] is,<ref name=ParrYangP247A.6>{{harv|Parr|Yang|1989|loc=p. 247, Eq. A.6}}.</ref>
:<math>
\begin{align}
\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}) \, .
\end{align}
</math>
 
====Coulomb potential energy functional====
For the '''electron-nucleus potential''', Thomas and Fermi employed the [[Coulomb's law|Coulomb]] potential energy functional
 
:<math>V[\rho] =  \int \frac{\rho(\boldsymbol{r})}{|\boldsymbol{r}|} \ d\boldsymbol{r}.</math>
 
Applying the definition of functional derivative,
 
:<math>
\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} \\
& {} =  \int  \frac {1} {|\boldsymbol{r}|} \, \phi(\boldsymbol{r}) \ d\boldsymbol{r} \, .
\end{align}
</math>
So,
:<math> \frac{\delta V}{\delta \rho(\boldsymbol{r})} = \frac{1}{|\boldsymbol{r}|} \ . </math>
 
For the classical part of the '''electron-electron interaction''', Thomas and Fermi employed the [[Coulomb's law|Coulomb]] potential energy functional
:<math>J[\rho] = \frac{1}{2}\iint \frac{\rho(\mathbf{r}) \rho(\mathbf{r}')}{\vert \mathbf{r}-\mathbf{r}' \vert}\, d\mathbf{r} d\mathbf{r}' \, .</math>
From the [[Functional_derivative#Functional_derivative|definition of the functional derivative]],
:<math>
\begin{align}
\int \frac{\delta J}{\delta\rho(\boldsymbol{r})} \phi(\boldsymbol{r})d\boldsymbol{r} 
& {} = \left [ \frac {d \ }{d\epsilon} \, J[\rho + \epsilon\phi] \right ]_{\epsilon = 0} \\
& {} = \left [ \frac {d \ }{d\epsilon} \, \left ( \frac{1}{2}\iint \frac {[\rho(\boldsymbol{r}) + \epsilon \phi(\boldsymbol{r})] \,  [\rho(\boldsymbol{r}') + \epsilon \phi(\boldsymbol{r}')]  }{\vert \boldsymbol{r}-\boldsymbol{r}' \vert}\, d\boldsymbol{r} d\boldsymbol{r}'  \right ) \right ]_{\epsilon = 0} \\
& {} =  \frac{1}{2}\iint \frac {\rho(\boldsymbol{r}') \phi(\boldsymbol{r})  }{\vert \boldsymbol{r}-\boldsymbol{r}' \vert}\, d\boldsymbol{r} d\boldsymbol{r}' +
            \frac{1}{2}\iint \frac {\rho(\boldsymbol{r}) \phi(\boldsymbol{r}')  }{\vert \boldsymbol{r}-\boldsymbol{r}' \vert}\, 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> \int \frac{\delta J}{\delta\rho(\boldsymbol{r})} \phi(\boldsymbol{r})d\boldsymbol{r} = \int \left ( \int \frac {\rho(\boldsymbol{r}') }{\vert \boldsymbol{r}-\boldsymbol{r}' \vert} 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>{{harv|Parr|Yang|1989|loc=p. 248, Eq. A.11}}.</ref>
:<math> \frac{\delta J}{\delta\rho(\boldsymbol{r})} = \int \frac {\rho(\boldsymbol{r}') }{\vert \boldsymbol{r}-\boldsymbol{r}' \vert} d\boldsymbol{r}' \, . </math>
 
The second functional derivative is
:<math>\frac{\delta^2 J[\rho]}{\delta \rho(\mathbf{r}')\delta\rho(\mathbf{r})}  = \frac{\partial}{\partial \rho(\mathbf{r}')} \left ( \frac{\rho(\mathbf{r}')}{\vert \mathbf{r}-\mathbf{r}' \vert} \right ) = \frac{1}{\vert \mathbf{r}-\mathbf{r}' \vert}.
</math>
 
====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 suit better a molecular electron cloud:
:<math>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} \ d\mathbf{r} \, ,</math>
where
:<math> 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 [[Functional_derivative#Formula|formula]] for the functional derivative,
:<math>
\begin{align}
\frac{\delta T_\mathrm{W}}{\delta \rho(\boldsymbol{r})}
& = \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>{{harv|Parr|Yang|1989|loc= p. 247, Eq. A.9}}.</ref>
:<math> \frac{\delta T_\mathrm{W}}{\delta \rho(\boldsymbol{r})} =  \ \ \, \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>
\begin{align}
H[p(x)] = -\sum_x p(x) \log p(x)
\end{align}
 
</math>
Thus,
 
:<math>
\begin{align}
\sum_x \frac{\delta H}{\delta p(x)} \, \phi(x)
& {} = \left[ \frac{d}{d\epsilon} H[p(x) + \epsilon\phi(x)] \right]_{\epsilon=0}\\
& {} = \left [- \, \frac{d}{d\varepsilon}  \sum_x \, [p(x) + \varepsilon\phi(x)] \ \log [p(x) + \varepsilon\phi(x)] \right]_{\varepsilon=0} \\
& {} = \displaystyle -\sum_x \, [1+\log p(x)] \ \phi(x) \, .
\end{align}
</math>
 
Thus,
 
:<math>
\frac{\delta H}{\delta p(x)} = -1-\log p(x).
</math>
 
==== Exponential ====
 
Let
:<math> F[\varphi(x)]= e^{\int \varphi(x) g(x)dx}.</math>
 
Using the delta function as a test function,
:<math>
\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> \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>\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>
\begin{align}
\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}').
\end{align}
</math>
 
==Using the delta function as a test function==
In physics, it's 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):
 
: <math>\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|''ϕ''}}, so one might think that it should hold also when {{math|''ϕ''}} is chosen to be a specific function such as the [[Dirac delta function|delta function]].  However, the latter is not a valid test function.
 
In the definition, the functional derivative describes how the functional <math>F[\varphi(x)]</math> changes as a result of a small change in the entire function <math>\varphi(x)</math>. The particular form of the change in <math>\varphi(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>\varphi(x)</math> is varied only in the point <math>y</math>. Except for this point, there is no variation in <math>\varphi(x)</math>.
 
==Notes==
{{Reflist|group=Note}}
 
==Footnotes==
{{reflist|29em}}
 
==References==
*{{cite book | last1=Courant | first1=Richard | authorlink1=Richard Courant | last2=Hilbert | first2=David | authorlink2=David Hilbert | title = Methods of Mathematical Physics | volume = Vol. I | edition = First English | ref=harv | 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
| last = Frigyik
| first = Béla A.
| author-link =
| last2 = Srivastava
| first2 = Santosh
| author2-link =
| last3 = Gupta
| first3 = Maya R.
| author3-link =
| 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
| language =
| url = https://www.ee.washington.edu/techsite/papers/documents/UWEETR-2008-0001.pdf
}}.
*{{Citation
| last = Gelfand
| first = 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
| origyear = 1963
| year = 2000
| edition = Revised English
| url = http://store.doverpublications.com/0486414485.html
| doi =
| id =
| isbn = 978-0486414485
| mr = 0160139
| zbl = 0127.05402
}}.
*{{Citation
| last = Giaquinta
| first = 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
| url =
| isbn = 3-540-50625-X
| mr = 1368401
| zbl = 0853.49001
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*{{Citation
| last = Greiner
| first = Walter
   | last2 = Reinhardt
| first2 = Joachim
| title = Field quantization
| place = Berlin–Heidelberg–New York
| publisher = Springer-Verlag
| series = With a foreword by D. A. Bromley
| volume =
| origyear =
| year = 1996
| edition =
| chapter = Section 2.3 – Functional derivatives
| chapterurl = http://www.physics.byu.edu/faculty/berrondo/wt752/functional%20derivative.pdf
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| pages = 36–38
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| zbl = 0844.00006
}}.
*{{cite book |first1=R. G.|last1=Parr|first2=W.|last2=Yang| title = [http://books.google.com/?id=mGOpScSIwU4C&printsec=frontcover&dq=Density-Functional+Theory+of+Atoms+and+Molecules&cd=1#v=onepage&q Density-Functional Theory of Atoms and Molecules] | chapter = Appendix A, Functionals | pages = 246–254 | publisher = Oxford University Press | year = 1989 |location=New York| url =  | ref=harv | isbn = 978-0195042795}}
 
{{Functional Analysis}}
 
[[Category:Differential calculus]]
[[Category:Topological vector spaces]]
[[Category:Differential operators]]
[[Category:Calculus of variations]]
[[Category:Variational analysis]]

Revision as of 10:25, 21 February 2014

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