Determining functional derivatives Functional derivative

1 determining functional derivatives

1.1 formula
1.2 examples

1.2.1 thomas–fermi kinetic energy functional
1.2.2 coulomb potential energy functional
1.2.3 weizsäcker kinetic energy functional
1.2.4 entropy
1.2.5 exponential
1.2.6 functional derivative of function
1.2.7 functional derivative of iterated function

determining functional derivatives

we give formula determine functional derivatives common class of functionals can written integral of function , derivatives. generalization of euler–lagrange equation: indeed, functional derivative introduced in physics within derivation of lagrange equation of second kind principle of least action in lagrangian mechanics (18th century). first 3 examples below taken density functional theory (20th century), fourth statistical mechanics (19th century).

formula

given functional

f
[
ρ
]
=
∫
f
(

r

,
ρ
(

r

)
,
∇
ρ
(

r

)
)

d

r

,


{\displaystyle f[\rho ]=\int f({\boldsymbol {r}},\rho ({\boldsymbol {r}}),\nabla \rho ({\boldsymbol {r}}))\,d{\boldsymbol {r}},}

and function ϕ(r) vanishes on boundary of region of integration, previous section definition,

∫



δ
f


δ
ρ
(

r

)




ϕ
(

r

)

d

r




=


[


d

d
ε



∫
f
(

r

,
ρ
+
ε
ϕ
,
∇
ρ
+
ε
∇
ϕ
)

d

r

]


ε
=
0








=
∫

(



∂
f


∂
ρ




ϕ
+



∂
f


∂
∇
ρ



⋅
∇
ϕ
)

d

r







=
∫

[



∂
f


∂
ρ




ϕ
+
∇
⋅

(



∂
f


∂
∇
ρ




ϕ
)

−

(
∇
⋅



∂
f


∂
∇
ρ



)

ϕ
]

d

r







=
∫

[



∂
f


∂
ρ




ϕ
−

(
∇
⋅



∂
f


∂
∇
ρ



)

ϕ
]

d

r







=
∫

(



∂
f


∂
ρ



−
∇
⋅



∂
f


∂
∇
ρ



)

ϕ
(

r

)
 
d

r


.






{\displaystyle {\begin{aligned}\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{aligned}}}

the second line obtained using total derivative, ∂f /∂∇ρ derivative of scalar respect vector. third line obtained use of product rule divergence. fourth line obtained using divergence theorem , condition ϕ=0 on boundary of region of integration. since ϕ arbitrary function, applying fundamental lemma of calculus of variations last line, functional derivative is

where ρ = ρ(r) , f = f (r, ρ, ∇ρ). formula case of functional form given f[ρ] @ beginning of section. other functional forms, definition of functional derivative can used starting point determination. (see example coulomb potential energy functional.)

the above equation functional derivative can generalized case includes higher dimensions , higher order derivatives. functional be,

f
[
ρ
(

r

)
]
=
∫
f
(

r

,
ρ
(

r

)
,
∇
ρ
(

r

)
,

∇

(
2
)


ρ
(

r

)
,
…
,

∇

(
n
)


ρ
(

r

)
)

d

r

,


{\displaystyle 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}},}

where vector r ∈ ℝ, , ∇ tensor n components partial derivative operators of order i,

[

∇

(
i
)


]



α

1



α

2


⋯

α

i




=



∂


i



∂

r


α

1




∂

r


α

2




⋯
∂

r


α

i










where



α

1


,

α

2


,
⋯
,

α

i


=
1
,
2
,
⋯
,
n
 
.


{\displaystyle \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\ .}

an analogous application of definition of functional derivative yields

δ
f
[
ρ
]


δ
ρ









=



∂
f


∂
ρ



−
∇
⋅



∂
f


∂
(
∇
ρ
)



+

∇

(
2
)


⋅



∂
f


∂

(

∇

(
2
)


ρ
)




+
⋯
+
(
−
1

)

n



∇

(
n
)


⋅



∂
f


∂

(

∇

(
n
)


ρ
)













=



∂
f


∂
ρ



+

∑

i
=
1


n


(
−
1

)

i



∇

(
i
)


⋅



∂
f


∂

(

∇

(
i
)


ρ
)




 
.






{\displaystyle {\begin{aligned}{\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{aligned}}}

in last 2 equations, n components of tensor






∂
f


∂

(

∇

(
i
)


ρ
)






{\displaystyle {\frac {\partial f}{\partial \left(\nabla ^{(i)}\rho \right)}}}

partial derivatives of f respect partial derivatives of ρ,

[



∂
f


∂

(

∇

(
i
)


ρ
)




]



α

1



α

2


⋯

α

i




=



∂
f


∂

ρ


α

1



α

2


⋯

α

i










where



ρ


α

1



α

2


⋯

α

i




≡




∂


i


ρ


∂

r


α

1





∂

r


α

2




⋯
∂

r


α

i







 
,


{\displaystyle \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}}}}\ ,}

and tensor scalar product is,

∇

(
i
)


⋅



∂
f


∂

(

∇

(
i
)


ρ
)




=

∑


α

1


,

α

2


,
⋯
,

α

i


=
1


n


 



∂


i



∂

r


α

1





∂

r


α

2




⋯
∂

r


α

i







 



∂
f


∂

ρ


α

1



α

2


⋯

α

i







 
.


{\displaystyle \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}}}}\ .}



examples
thomas–fermi kinetic energy functional

the thomas–fermi model of 1927 used kinetic energy functional noninteracting uniform electron gas in first attempt of density-functional theory of electronic structure:

t


t
f



[
ρ
]
=

c


f



∫

ρ

5

/

3


(

r

)

d

r


.


{\displaystyle t_{\mathrm {tf} }[\rho ]=c_{\mathrm {f} }\int \rho ^{5/3}(\mathbf {r} )\,d\mathbf {r} \,.}

since integrand of ttf[ρ] not involve derivatives of ρ(r), functional derivative of ttf[ρ] is,

δ

t


t
f





δ
ρ
(

r

)






=

c


f






∂

ρ

5

/

3


(

r

)


∂
ρ
(

r

)









=


5
3



c


f




ρ

2

/

3


(

r

)

.






{\displaystyle {\begin{aligned}{\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{aligned}}}



coulomb potential energy functional

for electron-nucleus potential, thomas , fermi employed coulomb potential energy functional

v
[
ρ
]
=
∫



ρ
(

r

)



|


r


|




 
d

r

.


{\displaystyle v[\rho ]=\int {\frac {\rho ({\boldsymbol {r}})}{|{\boldsymbol {r}}|}}\ d{\boldsymbol {r}}.}

applying definition of functional derivative,

∫



δ
v


δ
ρ
(

r

)



 
ϕ
(

r

)
 
d

r







=


[


d

d
ε



∫



ρ
(

r

)
+
ε
ϕ
(

r

)



|


r


|




 
d

r

]


ε
=
0











=
∫


1


|


r


|





ϕ
(

r

)
 
d

r


.






{\displaystyle {\begin{aligned}\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{aligned}}}

so,

δ
v


δ
ρ
(

r

)



=


1


|


r


|




 
.


{\displaystyle {\frac {\delta v}{\delta \rho ({\boldsymbol {r}})}}={\frac {1}{|{\boldsymbol {r}}|}}\ .}

for classical part of electron-electron interaction, thomas , fermi employed coulomb potential energy functional

j
[
ρ
]
=


1
2


∬



ρ
(

r

)
ρ
(


r

′

)


|

r

−


r

′

|




d

r

d


r

′


.


{\displaystyle 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} \,.}

from definition of functional derivative,

∫



δ
j


δ
ρ
(

r

)



ϕ
(

r

)
d

r







=


[



d
 


d
ϵ




j
[
ρ
+
ϵ
ϕ
]
]


ϵ
=
0











=


[



d
 


d
ϵ





(


1
2


∬



[
ρ
(

r

)
+
ϵ
ϕ
(

r

)
]

[
ρ
(


r

′

)
+
ϵ
ϕ
(


r

′

)
]


|

r

−


r

′

|




d

r

d


r

′

)

]


ϵ
=
0











=


1
2


∬



ρ
(


r

′

)
ϕ
(

r

)


|

r

−


r

′

|




d

r

d


r

′

+


1
2


∬



ρ
(

r

)
ϕ
(


r

′

)


|

r

−


r

′

|




d

r

d


r

′







{\displaystyle {\begin{aligned}\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{aligned}}}

the first , second terms on right hand side of last equation equal, since r , r′ in second term can interchanged without changing value of integral. therefore,

∫



δ
j


δ
ρ
(

r

)



ϕ
(

r

)
d

r

=
∫

(
∫



ρ
(


r

′

)


|

r

−


r

′

|



d


r

′

)

ϕ
(

r

)
d

r



{\displaystyle \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}}}

and functional derivative of electron-electron coulomb potential energy functional j[ρ] is,

δ
j


δ
ρ
(

r

)



=
∫



ρ
(


r

′

)


|

r

−


r

′

|



d


r

′


.


{\displaystyle {\frac {\delta j}{\delta \rho ({\boldsymbol {r}})}}=\int {\frac {\rho ({\boldsymbol {r}} )}{\vert {\boldsymbol {r}}-{\boldsymbol {r}} \vert }}d{\boldsymbol {r}} \,.}

the second functional derivative is

δ

2


j
[
ρ
]


δ
ρ
(


r

′

)
δ
ρ
(

r

)



=


∂

∂
ρ
(


r

′

)




(



ρ
(


r

′

)


|

r

−


r

′

|



)

=


1

|

r

−


r

′

|



.


{\displaystyle {\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 }}.}



weizsäcker kinetic energy functional

in 1935 von weizsäcker proposed add gradient correction thomas-fermi kinetic energy functional make suit better molecular electron cloud:

t


w



[
ρ
]
=


1
8


∫



∇
ρ
(

r

)
⋅
∇
ρ
(

r

)


ρ
(

r

)



d

r

=
∫

t


w



 
d

r


,


{\displaystyle 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} \,,}

where

t


w



≡


1
8





∇
ρ
⋅
∇
ρ

ρ




and

 
 
ρ
=
ρ
(

r

)
 
.


{\displaystyle t_{\mathrm {w} }\equiv {\frac {1}{8}}{\frac {\nabla \rho \cdot \nabla \rho }{\rho }}\qquad {\text{and}}\ \ \rho =\rho ({\boldsymbol {r}})\ .}

using derived formula functional derivative,

δ

t


w





δ
ρ
(

r

)






=



∂

t


w





∂
ρ



−
∇
⋅



∂

t


w





∂
∇
ρ









=
−


1
8





∇
ρ
⋅
∇
ρ


ρ

2




−

(


1
4






∇

2


ρ

ρ


−


1
4





∇
ρ
⋅
∇
ρ


ρ

2




)



where

 
 

∇

2


=
∇
⋅
∇
 
,






{\displaystyle {\begin{aligned}{\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{aligned}}}

and result is,

δ

t


w





δ
ρ
(

r

)



=
 
 



1
8





∇
ρ
⋅
∇
ρ


ρ

2




−


1
4






∇

2


ρ

ρ


 
.


{\displaystyle {\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 }}\ .}



entropy

the entropy of discrete random variable functional of probability mass function.

h
[
p
(
x
)
]
=
−

∑

x


p
(
x
)
log
⁡
p
(
x
)






{\displaystyle {\begin{aligned}h[p(x)]=-\sum _{x}p(x)\log p(x)\end{aligned}}}

thus,

∑

x





δ
h


δ
p
(
x
)




ϕ
(
x
)






=


[


d

d
ϵ



h
[
p
(
x
)
+
ϵ
ϕ
(
x
)
]
]


ϵ
=
0











=


[
−



d

d
ε




∑

x



[
p
(
x
)
+
ε
ϕ
(
x
)
]
 
log
⁡
[
p
(
x
)
+
ε
ϕ
(
x
)
]
]


ε
=
0











=

−

∑

x



[
1
+
log
⁡
p
(
x
)
]
 
ϕ
(
x
)

.







{\displaystyle {\begin{aligned}\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{aligned}}}

thus,

δ
h


δ
p
(
x
)



=
−
1
−
log
⁡
p
(
x
)
.


{\displaystyle {\frac {\delta h}{\delta p(x)}}=-1-\log p(x).}



exponential

let

f
[
φ
(
x
)
]
=

e

∫
φ
(
x
)
g
(
x
)
d
x


.


{\displaystyle f[\varphi (x)]=e^{\int \varphi (x)g(x)dx}.}

using delta function test function,

δ
f
[
φ
(
x
)
]


δ
φ
(
y
)









=

lim

ε
→
0





f
[
φ
(
x
)
+
ε
δ
(
x
−
y
)
]
−
f
[
φ
(
x
)
]

ε











=

lim

ε
→
0






e

∫
(
φ
(
x
)
+
ε
δ
(
x
−
y
)
)
g
(
x
)
d
x


−

e

∫
φ
(
x
)
g
(
x
)
d
x



ε











=

e

∫
φ
(
x
)
g
(
x
)
d
x



lim

ε
→
0






e

ε
∫
δ
(
x
−
y
)
g
(
x
)
d
x


−
1

ε











=

e

∫
φ
(
x
)
g
(
x
)
d
x



lim

ε
→
0






e

ε
g
(
y
)


−
1

ε











=

e

∫
φ
(
x
)
g
(
x
)
d
x


g
(
y
)
.






{\displaystyle {\begin{aligned}{\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{aligned}}}

thus,

δ
f
[
φ
(
x
)
]


δ
φ
(
y
)



=
g
(
y
)
f
[
φ
(
x
)
]
.


{\displaystyle {\frac {\delta f[\varphi (x)]}{\delta \varphi (y)}}=g(y)f[\varphi (x)].}

this particularly useful in calculating correlation functions partition function in quantum field theory.

functional derivative of function

a function can written in form of integral functional. example,

ρ
(

r

)
=
f
[
ρ
]
=
∫
ρ
(


r

′

)
δ
(

r

−


r

′

)

d


r

′

.


{\displaystyle \rho ({\boldsymbol {r}})=f[\rho ]=\int \rho ({\boldsymbol {r}} )\delta ({\boldsymbol {r}}-{\boldsymbol {r}} )\,d{\boldsymbol {r}} .}

since integrand not depend on derivatives of ρ, functional derivative of ρ(r) is,

δ
ρ
(

r

)


δ
ρ
(


r

′

)



≡



δ
f


δ
ρ
(


r

′

)






=



∂
 
 


∂
ρ
(


r

′

)




[
ρ
(


r

′

)
δ
(

r

−


r

′

)
]






=
δ
(

r

−


r

′

)
.






{\displaystyle {\begin{aligned}{\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{aligned}}}



functional derivative of iterated function

the functional derivative of iterated function



f
(
f
(
x
)
)


{\displaystyle f(f(x))}

given by:

δ
f
(
f
(
x
)
)


δ
f
(
y
)



=

f
′

(
f
(
x
)
)
δ
(
x
−
y
)
+
δ
(
f
(
x
)
−
y
)


{\displaystyle {\frac {\delta f(f(x))}{\delta f(y)}}=f (f(x))\delta (x-y)+\delta (f(x)-y)}

and

δ
f
(
f
(
f
(
x
)
)
)


δ
f
(
y
)



=

f
′

(
f
(
f
(
x
)
)
(

f
′

(
f
(
x
)
)
δ
(
x
−
y
)
+
δ
(
f
(
x
)
−
y
)
)
+
δ
(
f
(
f
(
x
)
)
−
y
)


{\displaystyle {\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)}

in general:

δ

f

n


(
x
)


δ
f
(
y
)



=

f
′

(

f

n
−
1


(
x
)
)



δ

f

n
−
1


(
x
)


δ
f
(
y
)



+
δ
(

f

n
−
1


(
x
)
−
y
)


{\displaystyle {\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)}

putting in n=0 gives:

δ

f

−
1


(
x
)


δ
f
(
y
)



=
−



δ
(

f

−
1


(
x
)
−
y
)



f
′

(

f

−
1


(
x
)
)





{\displaystyle {\frac {\delta f^{-1}(x)}{\delta f(y)}}=-{\frac {\delta (f^{-1}(x)-y)}{f (f^{-1}(x))}}}

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