亥姆霍兹分解

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。此定理以物理學家赫爾曼·馮·亥姆霍茲為名。^[10]

这意味着任何矢量场 $F$ ，都可以视为两个势场（純量勢 $φ$ 和向量勢 $A$ ）之和。

定理內容

假定 $F$ 為定義在有界區域 $V \subseteq R 3$ 裡的二次連續可微向量場，且 $S$ 為 $V$ 的包圍面，則 $F$ 可被分解成無旋度及無散度兩部份：^[11]

\mathbf{F}=-\boldsymbol{\nabla}\Phi+\boldsymbol{\nabla}\times\mathbf{A}

，

其中

{\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'}

{\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 $F$ 在無窮遠處消失的比 $1/r$ 快，則純量勢及向量勢的第二項為零，也就是說 ^[12]

\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'

\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'

推導

假定我們有一個向量函數 $\mathbf{F}\left(\mathbf{r}\right)$ ，且其旋度 $\boldsymbol{\nabla}\times\mathbf{F}$ 及散度 $\boldsymbol{\nabla}\cdot\mathbf{F}$ 已知。利用狄拉克δ函数可將函數改寫成

\delta\left(\mathbf{r}-\mathbf{r}'\right)=-\frac{1}{4\pi}\nabla^{2}\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}

，

\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

\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

\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]

=-\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 $\boldsymbol{\nabla}\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}=-\boldsymbol{\nabla}'\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}$ ，我們可將上式改寫成

\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，

\mathbf{a}\cdot\boldsymbol{\nabla}\psi=-\psi\left(\boldsymbol{\nabla}\cdot\mathbf{a}\right)+\boldsymbol{\nabla}\cdot\left(\psi\mathbf{a}\right)

\mathbf{a}\times\boldsymbol{\nabla}\psi=\psi\left(\boldsymbol{\nabla}\times\mathbf{a}\right)-\boldsymbol{\nabla}\times\left(\psi\mathbf{a}\right)

。

可得

{\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，方程式可改寫成

{\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]}

{\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] }

。

定義

{\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'}

{\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

\mathbf{F}=-\boldsymbol{\nabla}\Phi+\boldsymbol{\nabla}\times\mathbf{A}

利用傅利葉轉換做推導

（疑似有错误）將F改寫成傅利葉轉換的形式：

\vec{\mathbf{F}}(\vec{r}) = \iiint \vec{\mathbf{G}}(\vec{\omega}) e^{\displaystyle i \, \vec{\omega} \cdot \vec{r}} d\vec{\omega}

純量場的傅利葉轉換是一個純量場，向量場的傅利葉轉換是一個維度相同的向量場。現在考慮以下純量場及向量場：

{\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

\vec{\mathbf{G}}(\vec{\omega}) = - i \,\vec{\omega} \, G_\Phi(\vec{\omega})  + i \, \vec{\omega} \times \vec{\mathbf{G}}_\mathbf{A}(\vec{\omega})

{\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

Helmholtz theorem （MathWorld）

[1] On Helmholtz's Theorem in Finite Regions. By Jean Bladel. Midwestern Universities Research Association, 1958.

[2] Hermann von Helmholtz. Clarendon Press, 1906. By Leo Koenigsberger. p357

[3] An Elementary Course in the Integral Calculus. By Daniel Alexander Murray. American Book Company, 1898. p8.

[4] J. W. Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis, page 237, link from Internet Archive

[5] Electromagnetic theory, Volume 1. By Oliver Heaviside. "The Electrician" printing and publishing company, limited, 1893.

[6] Elements of the differential calculus. By Wesley Stoker Barker Woolhouse. Weale, 1854.

[7] 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。

[8] Vector Calculus: With Applications to Physics. By James Byrnie Shaw. D. Van Nostrand, 1922. p205.
参见：格林公式。

[9] A Treatise on the Integral Calculus, Volume 2. By Joseph Edwards. Chelsea Publishing Company, 1922.

[10] 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.

[11] 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).

[12] 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.

[11] Helmholtz' Theorem (PDF). University of Vermont. [2014-08-14].

[griffiths-12] David J. Griffiths, Introduction to Electrodynamics, Prentice-Hall, 1999, p. 556.

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