在物理学 和数学 中的向量分析 中,亥姆霍兹定理 ,[1] [2] 或称向量分析基本定理 ,[3] [4] [5] [6] [7] [8] [9] 指出对于任意足够光滑 、快速衰减的三维向量场 可分解为一个无旋向量场 和一个螺线向量场 的和,这个过程被称作亥姆霍兹分解 。此定理以物理学家赫尔曼·冯·亥姆霍兹 为名。[10]
这意味着任何矢量场 F ,都可以视为两个势场(标势 φ 和矢势 A )之和。
定理内容
假定 F 为定义在有界区域 V ⊆ R 3 里的二次连续可微向量场,且 S 为 V 的包围面,则 F 可被分解成无旋度 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'}
如果 V = R 3 ,且 F 在无穷远处消失的比
1
/
r
{\displaystyle 1/r}
快,则标势及矢势的第二项为零,也就是说
[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)}
,且其旋度
∇
×
F
{\displaystyle \boldsymbol{\nabla}\times\mathbf{F}}
及散度
∇
⋅
F
{\displaystyle \boldsymbol{\nabla}\cdot\mathbf{F}}
已知。利用狄拉克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|}}
,我们可将上式改写成
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]}
。
利用以下二等式,
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}) + i G_\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. 参见:流数法 。
↑ Vector Calculus: With Applications to Physics. By James Byrnie Shaw . D. Van Nostrand, 1922. p205. 参见: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" (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 ] .
↑ 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. 92–93
George 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