# 亥姆霍兹分解

## 定理內容

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

$\Phi\left(\mathbf{r}\right)=\frac{1}{4\pi}\int_{V}\frac{\boldsymbol{\nabla}'\cdot\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' -\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}'\cdot\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S'$

$\mathbf{A}\left(\mathbf{r}\right)=\frac{1}{4\pi}\int_{V}\frac{\boldsymbol{\nabla}'\times\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' -\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}'\times\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S'$

$\Phi\left(\mathbf{r}\right)=\frac{1}{4\pi}\int_{\text{all space}}\frac{\boldsymbol{\nabla}'\cdot\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'$

$\mathbf{A}\left(\mathbf{r}\right)=\frac{1}{4\pi}\int_{\text{all space}}\frac{\boldsymbol{\nabla}'\times\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'$

## 推導

$\delta\left(\mathbf{r}-\mathbf{r}'\right)=-\frac{1}{4\pi}\nabla^{2}\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}$
$\mathbf{F}\left(\mathbf{r}\right)=\int_{V}\mathbf{F}\left(\mathbf{r}'\right)\delta\left(\mathbf{r}-\mathbf{r}'\right)\mathrm{d}V'=\int_{V}\mathbf{F}\left(\mathbf{r}'\right)\left(-\frac{1}{4\pi}\nabla^{2}\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\right)\mathrm{d}V'=-\frac{1}{4\pi}\nabla^{2}\int_{V}\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'$

$\nabla^{2}\mathbf{a}=\boldsymbol{\nabla}\left(\boldsymbol{\nabla}\cdot\mathbf{a}\right)-\boldsymbol{\nabla}\times\left(\boldsymbol{\nabla}\times\mathbf{a}\right)$

$\mathbf{F}\left(\mathbf{r}\right)=-\frac{1}{4\pi}\left[\boldsymbol{\nabla}\left(\boldsymbol{\nabla}\cdot\int_{V}\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)-\boldsymbol{\nabla}\times\left(\boldsymbol{\nabla}\times\int_{V}\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)\right]$
$=-\frac{1}{4\pi}\left[\boldsymbol{\nabla}\left(\int_{V}\mathbf{F}\left(\mathbf{r}'\right)\cdot\boldsymbol{\nabla}\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)+\boldsymbol{\nabla}\times\left(\int_{V}\mathbf{F}\left(\mathbf{r}'\right)\times\boldsymbol{\nabla}\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)\right]$

$\mathbf{F}\left(\mathbf{r}\right)=-\frac{1}{4\pi}\left[-\boldsymbol{\nabla}\left(\int_{V}\mathbf{F}\left(\mathbf{r}'\right)\cdot\boldsymbol{\nabla}'\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)-\boldsymbol{\nabla}\times\left(\int_{V}\mathbf{F}\left(\mathbf{r}'\right)\times\boldsymbol{\nabla}'\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V'\right)\right]$

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

$\mathbf{F}\left(\mathbf{r}\right)=-\frac{1}{4\pi}\left[-\boldsymbol{\nabla}\left( -\int_{V}\frac{\boldsymbol{\nabla}'\cdot\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' +\int_{V}\boldsymbol{\nabla}'\cdot\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' \right)-\boldsymbol{\nabla}\times\left( \int_{V}\frac{\boldsymbol{\nabla}'\times\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' -\int_{V}\boldsymbol{\nabla}'\times\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' \right)\right]$

$\mathbf{F}\left(\mathbf{r}\right)=-\frac{1}{4\pi}\left[-\boldsymbol{\nabla}\left( -\int_{V}\frac{\boldsymbol{\nabla}'\cdot\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' +\oint_{S}\mathbf{\hat{n}}'\cdot\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S' \right)-\boldsymbol{\nabla}\times\left( \int_{V}\frac{\boldsymbol{\nabla}'\times\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' -\oint_{S}\mathbf{\hat{n}}'\times\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S' \right)\right]$
$= -\boldsymbol{\nabla}\left[\frac{1}{4\pi}\int_{V}\frac{\boldsymbol{\nabla}'\cdot\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' -\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}'\cdot\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S'\right] +\boldsymbol{\nabla}\times\left[\frac{1}{4\pi}\int_{V}\frac{\boldsymbol{\nabla}'\times\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' -\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}'\times\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S'\right]$

$\Phi\left(\mathbf{r}\right)\equiv\frac{1}{4\pi}\int_{V}\frac{\boldsymbol{\nabla}'\cdot\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' -\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}'\cdot\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S'$
$\mathbf{A}\left(\mathbf{r}\right)\equiv\frac{1}{4\pi}\int_{V}\frac{\boldsymbol{\nabla}'\times\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}V' -\frac{1}{4\pi}\oint_{S}\mathbf{\hat{n}}'\times\frac{\mathbf{F}\left(\mathbf{r}'\right)}{\left|\mathbf{r}-\mathbf{r}'\right|}\mathrm{d}S'$

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

### 利用傅利葉轉換做推導

F改寫成傅利葉轉換的形式：

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

$\begin{array}{lll} G_\Phi(\vec{\omega}) = i\, \frac{\displaystyle \vec{\mathbf{G}}(\vec{\omega}) \cdot \vec{\omega}}{||\vec{\omega}||^2} & \quad\quad & \vec{\mathbf{G}}_\mathbf{A}(\vec{\omega}) = i\, \vec{\omega} \times \left( \vec{\mathbf{G}}(\vec{\omega}) + i G_\Phi(\vec{\omega}) \, \vec{\omega} \right) \\ && \\ \Phi(\vec{r}) = \displaystyle \iiint G_\Phi(\vec{\omega}) e^{\displaystyle i \, \vec{\omega} \cdot \vec{r}} d\vec{\omega} & & \vec{\mathbf{A}}(\vec{r}) = \displaystyle \iiint \vec{\mathbf{G}}_\mathbf{A}(\vec{\omega}) e^{\displaystyle i \, \vec{\omega} \cdot \vec{r}} d\vec{\omega} \end{array}$

$\vec{\mathbf{G}}(\vec{\omega}) = - i \,\vec{\omega} \, G_\Phi(\vec{\omega}) + i \, \vec{\omega} \times \vec{\mathbf{G}}_\mathbf{A}(\vec{\omega})$
$\begin{array}{lll}\vec{\mathbf{F}}(\vec{r}) &=& \displaystyle - \iiint i \, \vec{\omega}\, G_\Phi(\vec{\omega}) \, e^{\displaystyle i \, \vec{\omega} \cdot \vec{r}} d\vec{\omega} + \iiint i \, \vec{\omega} \times \vec{\mathbf{G}}_\mathbf{A}(\vec{\omega}) e^{\displaystyle i \, \vec{\omega} \cdot \vec{r}} d\vec{\omega} \\ &=& - \boldsymbol{\nabla} \Phi(\vec{r}) + \boldsymbol{\nabla} \times \vec{\mathbf{A}}(\vec{r}) \end{array}$

## 注释

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：
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. ^ Helmholtz' Theorem. University of Vermont.
12. ^ 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.