Formula Functional derivative

-




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}}}}\ .}





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