定理内容
假定 F V ⊆ R 3 S V F 旋度 及无散度 两部分：[11]
F
=
−
∇
Φ
+
∇
×
A
{\displaystyle \mathbf {F} =-{\boldsymbol {\nabla }}\Phi +{\boldsymbol {\nabla }}\times \mathbf {A} }
其中
Φ
(
r
)
=
1
4
π
∫
V
∇
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
V
′
−
1
4
π
∮
S
n
^
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
S
′
{\displaystyle \Phi \left(\mathbf {r} \right)={\frac {1}{4\pi }}\int _{V}{\frac {{\boldsymbol {\nabla }}'\cdot \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\cdot {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} S'}
A
(
r
)
=
1
4
π
∫
V
∇
′
×
F
(
r
′
)
|
r
−
r
′
|
d
V
′
−
1
4
π
∮
S
n
^
′
×
F
(
r
′
)
|
r
−
r
′
|
d
S
′
{\displaystyle \mathbf {A} \left(\mathbf {r} \right)={\frac {1}{4\pi }}\int _{V}{\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\times {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} S'}
V = R 3 F
1
/
r
{\displaystyle 1/r}
[12]
Φ
(
r
)
=
1
4
π
∫
all space
∇
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
V
′
{\displaystyle \Phi \left(\mathbf {r} \right)={\frac {1}{4\pi }}\int _{\text{all space}}{\frac {{\boldsymbol {\nabla }}'\cdot \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'}
A
(
r
)
=
1
4
π
∫
all space
∇
′
×
F
(
r
′
)
|
r
−
r
′
|
d
V
′
{\displaystyle \mathbf {A} \left(\mathbf {r} \right)={\frac {1}{4\pi }}\int _{\text{all space}}{\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'}
推导
假定我们有一个向量函数
F
(
r
)
{\displaystyle \mathbf {F} \left(\mathbf {r} \right)}
∇
×
F
{\displaystyle {\boldsymbol {\nabla }}\times \mathbf {F} }
∇
⋅
F
{\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {F} }
狄拉克δ函数 可将函数改写成
δ
(
r
−
r
′
)
=
−
1
4
π
∇
2
1
|
r
−
r
′
|
{\displaystyle \delta \left(\mathbf {r} -\mathbf {r} '\right)=-{\frac {1}{4\pi }}\nabla ^{2}{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}}
F
(
r
)
=
∫
V
F
(
r
′
)
δ
(
r
−
r
′
)
d
V
′
=
∫
V
F
(
r
′
)
(
−
1
4
π
∇
2
1
|
r
−
r
′
|
)
d
V
′
=
−
1
4
π
∇
2
∫
V
F
(
r
′
)
|
r
−
r
′
|
d
V
′
{\displaystyle \mathbf {F} \left(\mathbf {r} \right)=\int _{V}\mathbf {F} \left(\mathbf {r} '\right)\delta \left(\mathbf {r} -\mathbf {r} '\right)\mathrm {d} V'=\int _{V}\mathbf {F} \left(\mathbf {r} '\right)\left(-{\frac {1}{4\pi }}\nabla ^{2}{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}\right)\mathrm {d} V'=-{\frac {1}{4\pi }}\nabla ^{2}\int _{V}{\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'}
利用以下等式
∇
2
a
=
∇
(
∇
⋅
a
)
−
∇
×
(
∇
×
a
)
{\displaystyle \nabla ^{2}\mathbf {a} ={\boldsymbol {\nabla }}\left({\boldsymbol {\nabla }}\cdot \mathbf {a} \right)-{\boldsymbol {\nabla }}\times \left({\boldsymbol {\nabla }}\times \mathbf {a} \right)}
可得
F
(
r
)
=
−
1
4
π
[
∇
(
∇
⋅
∫
V
F
(
r
′
)
|
r
−
r
′
|
d
V
′
)
−
∇
×
(
∇
×
∫
V
F
(
r
′
)
|
r
−
r
′
|
d
V
′
)
]
{\displaystyle \mathbf {F} \left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\left[{\boldsymbol {\nabla }}\left({\boldsymbol {\nabla }}\cdot \int _{V}{\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'\right)-{\boldsymbol {\nabla }}\times \left({\boldsymbol {\nabla }}\times \int _{V}{\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'\right)\right]}
=
−
1
4
π
[
∇
(
∫
V
F
(
r
′
)
⋅
∇
1
|
r
−
r
′
|
d
V
′
)
+
∇
×
(
∫
V
F
(
r
′
)
×
∇
1
|
r
−
r
′
|
d
V
′
)
]
{\displaystyle =-{\frac {1}{4\pi }}\left[{\boldsymbol {\nabla }}\left(\int _{V}\mathbf {F} \left(\mathbf {r} '\right)\cdot {\boldsymbol {\nabla }}{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'\right)+{\boldsymbol {\nabla }}\times \left(\int _{V}\mathbf {F} \left(\mathbf {r} '\right)\times {\boldsymbol {\nabla }}{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'\right)\right]}
注意到
∇
1
|
r
−
r
′
|
=
−
∇
′
1
|
r
−
r
′
|
{\displaystyle {\boldsymbol {\nabla }}{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}=-{\boldsymbol {\nabla }}'{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}}
F
(
r
)
=
−
1
4
π
[
−
∇
(
∫
V
F
(
r
′
)
⋅
∇
′
1
|
r
−
r
′
|
d
V
′
)
−
∇
×
(
∫
V
F
(
r
′
)
×
∇
′
1
|
r
−
r
′
|
d
V
′
)
]
{\displaystyle \mathbf {F} \left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\left[-{\boldsymbol {\nabla }}\left(\int _{V}\mathbf {F} \left(\mathbf {r} '\right)\cdot {\boldsymbol {\nabla }}'{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'\right)-{\boldsymbol {\nabla }}\times \left(\int _{V}\mathbf {F} \left(\mathbf {r} '\right)\times {\boldsymbol {\nabla }}'{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'\right)\right]}
a
⋅
∇
ψ
=
−
ψ
(
∇
⋅
a
)
+
∇
⋅
(
ψ
a
)
{\displaystyle \mathbf {a} \cdot {\boldsymbol {\nabla }}\psi =-\psi \left({\boldsymbol {\nabla }}\cdot \mathbf {a} \right)+{\boldsymbol {\nabla }}\cdot \left(\psi \mathbf {a} \right)}
a
×
∇
ψ
=
ψ
(
∇
×
a
)
−
∇
×
(
ψ
a
)
{\displaystyle \mathbf {a} \times {\boldsymbol {\nabla }}\psi =\psi \left({\boldsymbol {\nabla }}\times \mathbf {a} \right)-{\boldsymbol {\nabla }}\times \left(\psi \mathbf {a} \right)}
可得
F
(
r
)
=
−
1
4
π
[
−
∇
(
−
∫
V
∇
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
V
′
+
∫
V
∇
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
V
′
)
−
∇
×
(
∫
V
∇
′
×
F
(
r
′
)
|
r
−
r
′
|
d
V
′
−
∫
V
∇
′
×
F
(
r
′
)
|
r
−
r
′
|
d
V
′
)
]
{\displaystyle \mathbf {F} \left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\left[-{\boldsymbol {\nabla }}\left(-\int _{V}{\frac {{\boldsymbol {\nabla }}'\cdot \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'+\int _{V}{\boldsymbol {\nabla }}'\cdot {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'\right)-{\boldsymbol {\nabla }}\times \left(\int _{V}{\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'-\int _{V}{\boldsymbol {\nabla }}'\times {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'\right)\right]}
利用散度定理 ，方程可改写成
F
(
r
)
=
−
1
4
π
[
−
∇
(
−
∫
V
∇
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
V
′
+
∮
S
n
^
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
S
′
)
−
∇
×
(
∫
V
∇
′
×
F
(
r
′
)
|
r
−
r
′
|
d
V
′
−
∮
S
n
^
′
×
F
(
r
′
)
|
r
−
r
′
|
d
S
′
)
]
{\displaystyle \mathbf {F} \left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\left[-{\boldsymbol {\nabla }}\left(-\int _{V}{\frac {{\boldsymbol {\nabla }}'\cdot \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'+\oint _{S}\mathbf {\hat {n}} '\cdot {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} S'\right)-{\boldsymbol {\nabla }}\times \left(\int _{V}{\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'-\oint _{S}\mathbf {\hat {n}} '\times {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} S'\right)\right]}
=
−
∇
[
1
4
π
∫
V
∇
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
V
′
−
1
4
π
∮
S
n
^
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
S
′
]
+
∇
×
[
1
4
π
∫
V
∇
′
×
F
(
r
′
)
|
r
−
r
′
|
d
V
′
−
1
4
π
∮
S
n
^
′
×
F
(
r
′
)
|
r
−
r
′
|
d
S
′
]
{\displaystyle =-{\boldsymbol {\nabla }}\left[{\frac {1}{4\pi }}\int _{V}{\frac {{\boldsymbol {\nabla }}'\cdot \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\cdot {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} S'\right]+{\boldsymbol {\nabla }}\times \left[{\frac {1}{4\pi }}\int _{V}{\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\times {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} S'\right]}
定义
Φ
(
r
)
≡
1
4
π
∫
V
∇
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
V
′
−
1
4
π
∮
S
n
^
′
⋅
F
(
r
′
)
|
r
−
r
′
|
d
S
′
{\displaystyle \Phi \left(\mathbf {r} \right)\equiv {\frac {1}{4\pi }}\int _{V}{\frac {{\boldsymbol {\nabla }}'\cdot \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\cdot {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} S'}
A
(
r
)
≡
1
4
π
∫
V
∇
′
×
F
(
r
′
)
|
r
−
r
′
|
d
V
′
−
1
4
π
∮
S
n
^
′
×
F
(
r
′
)
|
r
−
r
′
|
d
S
′
{\displaystyle \mathbf {A} \left(\mathbf {r} \right)\equiv {\frac {1}{4\pi }}\int _{V}{\frac {{\boldsymbol {\nabla }}'\times \mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\times {\frac {\mathbf {F} \left(\mathbf {r} '\right)}{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} S'}
所以
F
=
−
∇
Φ
+
∇
×
A
{\displaystyle \mathbf {F} =-{\boldsymbol {\nabla }}\Phi +{\boldsymbol {\nabla }}\times \mathbf {A} }
利用傅里叶变换做推导 （疑似有错误）
将F 改写成傅里叶变换 的形式：
F
→
(
r
→
)
=
∭
G
→
(
ω
→
)
e
i
ω
→
⋅
r
→
d
ω
→
{\displaystyle {\vec {\mathbf {F} }}({\vec {r}})=\iiint {\vec {\mathbf {G} }}({\vec {\omega }})e^{\displaystyle i\,{\vec {\omega }}\cdot {\vec {r}}}d{\vec {\omega }}}
标量场的傅里叶变换是一个标量场，向量场的傅里叶变换是一个维度相同的向量场。
现在考虑以下标量场及向量场：
G
Φ
(
ω
→
)
=
i
G
→
(
ω
→
)
⋅
ω
→
|
|
ω
→
|
|
2
G
→
A
(
ω
→
)
=
i
ω
→
×
(
G
→
(
ω
→
)
+
i
G
Φ
(
ω
→
)
ω
→
)
Φ
(
r
→
)
=
∭
G
Φ
(
ω
→
)
e
i
ω
→
⋅
r
→
d
ω
→
A
→
(
r
→
)
=
∭
G
→
A
(
ω
→
)
e
i
ω
→
⋅
r
→
d
ω
→
{\displaystyle {\begin{array}{lll}G_{\Phi }({\vec {\omega }})=i\,{\frac {\displaystyle {\vec {\mathbf {G} }}({\vec {\omega }})\cdot {\vec {\omega }}}{||{\vec {\omega }}||^{2}}}&\quad \quad &{\vec {\mathbf {G} }}_{\mathbf {A} }({\vec {\omega }})=i\,{\vec {\omega }}\times \left({\vec {\mathbf {G} }}({\vec {\omega }})+iG_{\Phi }({\vec {\omega }})\,{\vec {\omega }}\right)\\&&\\\Phi ({\vec {r}})=\displaystyle \iiint G_{\Phi }({\vec {\omega }})e^{\displaystyle i\,{\vec {\omega }}\cdot {\vec {r}}}d{\vec {\omega }}&&{\vec {\mathbf {A} }}({\vec {r}})=\displaystyle \iiint {\vec {\mathbf {G} }}_{\mathbf {A} }({\vec {\omega }})e^{\displaystyle i\,{\vec {\omega }}\cdot {\vec {r}}}d{\vec {\omega }}\end{array}}}
所以
G
→
(
ω
→
)
=
−
i
ω
→
G
Φ
(
ω
→
)
+
i
ω
→
×
G
→
A
(
ω
→
)
{\displaystyle {\vec {\mathbf {G} }}({\vec {\omega }})=-i\,{\vec {\omega }}\,G_{\Phi }({\vec {\omega }})+i\,{\vec {\omega }}\times {\vec {\mathbf {G} }}_{\mathbf {A} }({\vec {\omega }})}
F
→
(
r
→
)
=
−
∭
i
ω
→
G
Φ
(
ω
→
)
e
i
ω
→
⋅
r
→
d
ω
→
+
∭
i
ω
→
×
G
→
A
(
ω
→
)
e
i
ω
→
⋅
r
→
d
ω
→
=
−
∇
Φ
(
r
→
)
+
∇
×
A
→
(
r
→
)
{\displaystyle {\begin{array}{lll}{\vec {\mathbf {F} }}({\vec {r}})&=&\displaystyle -\iiint i\,{\vec {\omega }}\,G_{\Phi }({\vec {\omega }})\,e^{\displaystyle i\,{\vec {\omega }}\cdot {\vec {r}}}d{\vec {\omega }}+\iiint i\,{\vec {\omega }}\times {\vec {\mathbf {G} }}_{\mathbf {A} }({\vec {\omega }})e^{\displaystyle i\,{\vec {\omega }}\cdot {\vec {r}}}d{\vec {\omega }}\\&=&-{\boldsymbol {\nabla }}\Phi ({\vec {r}})+{\boldsymbol {\nabla }}\times {\vec {\mathbf {A} }}({\vec {r}})\end{array}}}
注释
^ On Helmholtz's Theorem in Finite Regions. By Jean Bladel . Midwestern Universities Research Association, 1958.
^ Hermann von Helmholtz. Clarendon Press, 1906. By Leo Koenigsberger . p357
^ An Elementary Course in the Integral Calculus. By Daniel Alexander Murray . American Book Company, 1898. p8.
^ J. W. Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis , page 237, link from Internet Archive ^ Electromagnetic theory, Volume 1. By Oliver Heaviside . "The Electrician" printing and publishing company, limited, 1893.
^ Elements of the differential calculus. By Wesley Stoker Barker Woolhouse . Weale, 1854.
^ An Elementary Treatise on the Integral Calculus: Founded on the Method of Rates Or Fluxions. By William Woolsey Johnson . John Wiley & Sons, 1881.流数法 。
^ Vector Calculus: With Applications to Physics. By James Byrnie Shaw . D. Van Nostrand, 1922. p205.格林公式 。
^ A Treatise on the Integral Calculus, Volume 2. By Joseph Edwards . Chelsea Publishing Company, 1922.
^ 参见：
H. Helmholtz (1858) "Über Integrale der hydrodynamischen Gleichungen, welcher der Wirbelbewegungen entsprechen" （页面存档备份 ，存于互联网档案馆 ） (On integrals of the hydrodynamic equations which correspond to vortex motions), Journal für die reine und angewandte Mathematik , 55 : 25-55. On page 38, the components of the fluid's velocity (u, v, w) are expressed in terms of the gradient of a scalar potential P and the curl of a vector potential (L, M, N).
However, Helmholtz was largely anticipated by George Stokes in his paper: G. G. Stokes (presented: 1849 ; published: 1856) "On the dynamical theory of diffraction," Transactions of the Cambridge Philosophical Society , vol. 9, part I, pages 1-62; see pages 9-10. ^ Helmholtz' Theorem (PDF) . University of Vermont. [2014-08-14 ] . （原始内容 (PDF) 存档于2012-08-13）. ^ David J. Griffiths, Introduction to Electrodynamics , Prentice-Hall, 1999, p. 556.
参考文献
一般参考文献 George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists, 4th edition, Academic Press: San Diego (1995) pp. 92–93George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists International Edition, 6th edition, Academic Press: San Diego (2005) pp. 95–101 弱形式的参考文献 C. Amrouche, C. Bernardi, M. Dauge, and V. Girault. "Vector potentials in three dimensional non-smooth domains." Mathematical Methods in the Applied Sciences , 21 , 823–864, 1998.
R. Dautray and J.-L. Lions. Spectral Theory and Applications, volume 3 of Mathematical Analysis and Numerical Methods for Science and Technology. Springer-Verlag, 1990.
V. Girault and P.A. Raviart. Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, 1986. 外部链接