Definition Functional derivative

1 definition

1.1 functional derivative
1.2 functional differential
1.3 formal description

definition

in section, functional derivative defined. functional differential defined in terms of functional derivative.

functional derivative

given manifold m representing (continuous/smooth) functions ρ (with boundary conditions etc.), , functional f defined as

f
:
m
→

r



or


f
:
m
→

c


,


{\displaystyle f\colon m\rightarrow \mathbb {r} \quad {\mbox{or}}\quad f\colon m\rightarrow \mathbb {c} \,,}

the functional derivative of f[ρ], denoted δf/δρ, defined by

∫



δ
f


δ
ρ



(
x
)
ϕ
(
x
)

d
x



=

lim

ε
→
0





f
[
ρ
+
ε
ϕ
]
−
f
[
ρ
]

ε








=


[


d

d
ϵ



f
[
ρ
+
ϵ
ϕ
]
]


ϵ
=
0


,






{\displaystyle {\begin{aligned}\int {\frac {\delta f}{\delta \rho }}(x)\phi (x)\;dx&=\lim _{\varepsilon \to 0}{\frac {f[\rho +\varepsilon \phi ]-f[\rho ]}{\varepsilon }}\\&=\left[{\frac {d}{d\epsilon }}f[\rho +\epsilon \phi ]\right]_{\epsilon =0},\end{aligned}}}

where



ϕ


{\displaystyle \phi }

arbitrary function. quantity εϕ called variation of ρ. in other words,

ϕ
↦


[


d

d
ϵ



f
[
ρ
+
ϵ
ϕ
]
]


ϵ
=
0




{\displaystyle \phi \mapsto \left[{\frac {d}{d\epsilon }}f[\rho +\epsilon \phi ]\right]_{\epsilon =0}}

is linear functional, riesz–markov–kakutani representation theorem, functional given integration against measure. δf/δρ defined radon–nikodym derivative of measure.

we think of function δf/δρ gradient of f @ point ρ and

∫



δ
f


δ
ρ



(
x
)
ϕ
(
x
)

d
x


{\displaystyle \int {\frac {\delta f}{\delta \rho }}(x)\phi (x)\;dx}

as directional derivative @ point ρ in direction of ϕ. analogous vector calculus, inner product gradient gives directional derivative.

functional differential

the differential (or variation or first variation) of functional f[ρ]

δ
f
(
ρ
,
ϕ
)
=
∫



δ
f


δ
ρ



(
x
)
 
ϕ
(
x
)
 
d
x
 
.


{\displaystyle \delta f(\rho ,\phi )=\int {\frac {\delta f}{\delta \rho }}(x)\ \phi (x)\ dx\ .}

heuristically, ϕ change in ρ, formally have ϕ = δρ, , similar in form total differential of function f(ρ1, ρ2, ..., ρn),

d
f
=

∑

i
=
1


n





∂
f


∂

ρ

i





 
d

ρ

i


 
,


{\displaystyle df=\sum _{i=1}^{n}{\frac {\partial f}{\partial \rho _{i}}}\ d\rho _{i}\ ,}

where ρ1, ρ2, ... , ρn independent variables. comparing last 2 equations, functional derivative δf/δρ(x) has role similar of partial derivative ∂f/∂ρi , variable of integration x continuous version of summation index i.

formal description

the definition of functional derivative may made more mathematically precise , formal defining space of functions more carefully. example, when space of functions banach space, functional derivative becomes known fréchet derivative, while 1 uses gâteaux derivative on more general locally convex spaces. note hilbert spaces special cases of banach spaces. more formal treatment allows many theorems ordinary calculus , analysis generalized corresponding theorems in functional analysis, numerous new theorems stated.

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