The Helmholtz Decomposition Theorem, regarded as the fundamental theorem of vector calculus, dictates that any vector field
F
{\displaystyle \mathbf {F} }
can be expressed as the sum of a conservative vector field
G
{\displaystyle \mathbf {G} }
and a divergence free vector field
H
{\displaystyle \mathbf {H} }
:
F
=
G
+
H
{\displaystyle \mathbf {F} =\mathbf {G} +\mathbf {H} }
.
Given a vector field
F
{\displaystyle \mathbf {F} }
, the vector field
G
(
q
)
=
1
4
π
∭
q
′
∈
R
3
(
∇
⋅
F
)
|
q
′
(
q
−
q
′
)
|
q
−
q
′
|
3
d
V
′
{\displaystyle \mathbf {G} (\mathbf {q} )={\frac {1}{4\pi }}\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {(\nabla \cdot \mathbf {F} )|_{\mathbf {q} '}(\mathbf {q} -\mathbf {q} ')}{|\mathbf {q} -\mathbf {q} '|^{3}}}dV'}
has the same divergence as
F
{\displaystyle \mathbf {F} }
, and is also conservative:
∇
⋅
G
=
∇
⋅
F
{\displaystyle \nabla \cdot \mathbf {G} =\nabla \cdot \mathbf {F} }
and
∇
×
G
=
0
{\displaystyle \nabla \times \mathbf {G} =\mathbf {0} }
. The vector field
H
=
F
−
G
{\displaystyle \mathbf {H} =\mathbf {F} -\mathbf {G} }
is divergence free.
Therefore
F
=
G
+
H
{\displaystyle \mathbf {F} =\mathbf {G} +\mathbf {H} }
where
G
(
q
)
=
1
4
π
∭
q
′
∈
R
3
(
∇
⋅
F
)
|
q
′
(
q
−
q
′
)
|
q
−
q
′
|
3
d
V
′
{\displaystyle \mathbf {G} (\mathbf {q} )={\frac {1}{4\pi }}\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {(\nabla \cdot \mathbf {F} )|_{\mathbf {q} '}(\mathbf {q} -\mathbf {q} ')}{|\mathbf {q} -\mathbf {q} '|^{3}}}dV'}
and
H
=
F
−
G
{\displaystyle \mathbf {H} =\mathbf {F} -\mathbf {G} }
. Vector field
G
{\displaystyle \mathbf {G} }
is conservative and
H
{\displaystyle \mathbf {H} }
is divergence free.
Given a vector field
F
{\displaystyle \mathbf {F} }
, the vector field
H
(
q
)
=
1
4
π
∭
q
′
∈
R
3
(
∇
×
F
)
|
q
′
×
(
q
−
q
′
)
|
q
−
q
′
|
3
d
V
′
{\displaystyle \mathbf {H} (\mathbf {q} )={\frac {1}{4\pi }}\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {(\nabla \times \mathbf {F} )|_{\mathbf {q} '}\times (\mathbf {q} -\mathbf {q} ')}{|\mathbf {q} -\mathbf {q} '|^{3}}}dV'}
has the same curl as
F
{\displaystyle \mathbf {F} }
, and is also divergence free:
∇
×
H
=
∇
×
F
{\displaystyle \nabla \times \mathbf {H} =\nabla \times \mathbf {F} }
and
∇
⋅
H
=
0
{\displaystyle \nabla \cdot \mathbf {H} =0}
. The vector field
G
=
F
−
H
{\displaystyle \mathbf {G} =\mathbf {F} -\mathbf {H} }
is conservative.
Therefore
F
=
G
+
H
{\displaystyle \mathbf {F} =\mathbf {G} +\mathbf {H} }
where
H
(
q
)
=
1
4
π
∭
q
′
∈
R
3
(
∇
×
F
)
|
q
′
×
(
q
−
q
′
)
|
q
−
q
′
|
3
d
V
′
{\displaystyle \mathbf {H} (\mathbf {q} )={\frac {1}{4\pi }}\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {(\nabla \times \mathbf {F} )|_{\mathbf {q} '}\times (\mathbf {q} -\mathbf {q} ')}{|\mathbf {q} -\mathbf {q} '|^{3}}}dV'}
and
G
=
F
−
H
{\displaystyle \mathbf {G} =\mathbf {F} -\mathbf {H} }
. Vector field
G
{\displaystyle \mathbf {G} }
is conservative and
H
{\displaystyle \mathbf {H} }
is divergence free.
The Helmholtz decomposition can be derived as follows:
Given an arbitrary point
q
′
{\displaystyle \mathbf {q} '}
, the divergence of the vector field
q
−
q
′
4
π
|
q
−
q
′
|
3
{\displaystyle {\frac {\mathbf {q} -\mathbf {q} '}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}}
is
∇
q
⋅
q
−
q
′
4
π
|
q
−
q
′
|
3
=
δ
(
q
;
q
′
)
{\displaystyle \nabla _{\mathbf {q} }\cdot {\frac {\mathbf {q} -\mathbf {q} '}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}=\delta (\mathbf {q} ;\mathbf {q} ')}
where
δ
(
q
;
q
′
)
{\displaystyle \delta (\mathbf {q} ;\mathbf {q} ')}
is the Dirac delta function centered on
q
′
{\displaystyle \mathbf {q} '}
(The subscript
q
{\displaystyle _{\mathbf {q} }}
clarifies that
q
{\displaystyle \mathbf {q} }
as opposed to
q
′
{\displaystyle \mathbf {q} '}
is the parameter that the differential operator is being applied to). Since
∇
q
(
−
1
|
q
−
q
′
|
)
=
q
−
q
′
|
q
−
q
′
|
3
{\displaystyle \nabla _{\mathbf {q} }({\frac {-1}{|\mathbf {q} -\mathbf {q} '|}})={\frac {\mathbf {q} -\mathbf {q} '}{|\mathbf {q} -\mathbf {q} '|^{3}}}}
, it is the case that
∇
q
2
−
1
4
π
|
q
−
q
′
|
=
∇
q
⋅
q
−
q
′
4
π
|
q
−
q
′
|
3
=
δ
(
q
;
q
′
)
{\displaystyle \nabla _{\mathbf {q} }^{2}{\frac {-1}{4\pi |\mathbf {q} -\mathbf {q} '|}}=\nabla _{\mathbf {q} }\cdot {\frac {\mathbf {q} -\mathbf {q} '}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}=\delta (\mathbf {q} ;\mathbf {q} ')}
Alongside the identities
∇
⋅
(
f
G
)
=
(
∇
f
)
⋅
G
+
f
(
∇
⋅
G
)
{\displaystyle \nabla \cdot (f\mathbf {G} )=(\nabla f)\cdot \mathbf {G} +f(\nabla \cdot \mathbf {G} )}
, and
∇
×
(
f
G
)
=
(
∇
f
)
×
G
+
f
(
∇
×
G
)
{\displaystyle \nabla \times (f\mathbf {G} )=(\nabla f)\times \mathbf {G} +f(\nabla \times \mathbf {G} )}
, and most importantly
∇
×
(
∇
×
F
)
=
∇
(
∇
⋅
F
)
−
∇
2
F
{\displaystyle \nabla \times (\nabla \times \mathbf {F} )=\nabla (\nabla \cdot \mathbf {F} )-\nabla ^{2}\mathbf {F} }
, the following can be derived:
F
(
q
)
=
∭
q
′
∈
R
3
δ
(
q
;
q
′
)
F
(
q
′
)
d
V
′
{\displaystyle \mathbf {F} (\mathbf {q} )=\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}\delta (\mathbf {q} ;\mathbf {q} ')\mathbf {F} (\mathbf {q} ')dV'}
=
∭
q
′
∈
R
3
(
∇
q
2
−
1
4
π
|
q
−
q
′
|
)
F
(
q
′
)
d
V
′
{\displaystyle =\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}(\nabla _{\mathbf {q} }^{2}{\frac {-1}{4\pi |\mathbf {q} -\mathbf {q} '|}})\mathbf {F} (\mathbf {q} ')dV'}
=
∭
q
′
∈
R
3
(
∇
q
2
−
F
(
q
′
)
4
π
|
q
−
q
′
|
)
d
V
′
{\displaystyle =\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}(\nabla _{\mathbf {q} }^{2}{\frac {-\mathbf {F} (\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|}})dV'}
=
∭
q
′
∈
R
3
(
∇
q
(
∇
q
⋅
−
F
(
q
′
)
4
π
|
q
−
q
′
|
)
−
∇
q
×
(
∇
q
×
−
F
(
q
′
)
4
π
|
q
−
q
′
|
)
)
d
V
′
{\displaystyle =\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}(\nabla _{\mathbf {q} }(\nabla _{\mathbf {q} }\cdot {\frac {-\mathbf {F} (\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|}})-\nabla _{\mathbf {q} }\times (\nabla _{\mathbf {q} }\times {\frac {-\mathbf {F} (\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|}}))dV'}
=
∭
q
′
∈
R
3
(
∇
q
(
(
q
−
q
′
)
⋅
F
(
q
′
)
4
π
|
q
−
q
′
|
3
)
−
∇
q
×
(
(
q
−
q
′
)
×
F
(
q
′
)
4
π
|
q
−
q
′
|
3
)
)
d
V
′
{\displaystyle =\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}(\nabla _{\mathbf {q} }({\frac {(\mathbf {q} -\mathbf {q} ')\cdot \mathbf {F} (\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}})-\nabla _{\mathbf {q} }\times ({\frac {(\mathbf {q} -\mathbf {q} ')\times \mathbf {F} (\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}))dV'}
=
∇
q
∭
q
′
∈
R
3
F
(
q
′
)
⋅
(
q
−
q
′
)
4
π
|
q
−
q
′
|
3
d
V
′
+
∇
q
×
∭
q
′
∈
R
3
F
(
q
′
)
×
(
q
−
q
′
)
4
π
|
q
−
q
′
|
3
d
V
′
{\displaystyle =\nabla _{\mathbf {q} }\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\cdot (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'+\nabla _{\mathbf {q} }\times \iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\times (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'}
G
(
q
)
=
∇
q
∭
q
′
∈
R
3
F
(
q
′
)
⋅
(
q
−
q
′
)
4
π
|
q
−
q
′
|
3
d
V
′
{\displaystyle \mathbf {G} (\mathbf {q} )=\nabla _{\mathbf {q} }\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\cdot (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'}
is the gradient of a scalar field, and so is conservative.
H
(
q
)
=
∇
q
×
∭
q
′
∈
R
3
F
(
q
′
)
×
(
q
−
q
′
)
4
π
|
q
−
q
′
|
3
d
V
′
{\displaystyle \mathbf {H} (\mathbf {q} )=\nabla _{\mathbf {q} }\times \iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\times (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'}
is the curl of a vector field, and so is divergence free.
In summary,
F
=
G
+
H
{\displaystyle \mathbf {F} =\mathbf {G} +\mathbf {H} }
where
G
(
q
)
=
∇
q
∭
q
′
∈
R
3
F
(
q
′
)
⋅
(
q
−
q
′
)
4
π
|
q
−
q
′
|
3
d
V
′
{\displaystyle \mathbf {G} (\mathbf {q} )=\nabla _{\mathbf {q} }\iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\cdot (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'}
is conservative and
H
(
q
)
=
∇
q
×
∭
q
′
∈
R
3
F
(
q
′
)
×
(
q
−
q
′
)
4
π
|
q
−
q
′
|
3
d
V
′
{\displaystyle \mathbf {H} (\mathbf {q} )=\nabla _{\mathbf {q} }\times \iiint _{\mathbf {q} '\in \mathbb {R} ^{3}}{\frac {\mathbf {F} (\mathbf {q} ')\times (\mathbf {q} -\mathbf {q} ')}{4\pi |\mathbf {q} -\mathbf {q} '|^{3}}}dV'}
is divergence free.