查看“︁亥姆霍兹分解”︁的源代码

{{NoteTA
|G1=Physics
|G2=Math
}}

在[[物理学]]和[[数学]]中的[[向量分析]]中，'''亥姆霍兹定理'''，<ref>On Helmholtz's Theorem in Finite Regions. By [[Jean Bladel]]. Midwestern Universities Research Association, 1958.</ref><ref>Hermann von Helmholtz. Clarendon Press, 1906. By [[Leo Koenigsberger]]. p357</ref> 或称'''向量分析基本定理'''，<ref>An Elementary Course in the Integral Calculus. By [[Daniel Alexander Murray]]. American Book Company, 1898. p8.</ref><ref>[[约西亚·吉布斯|J. W. Gibbs]] & [[Edwin Bidwell Wilson]] (1901) [https://archive.org/stream/117714283#page/236/mode/2up Vector Analysis], page 237, link from [[互联网档案馆|Internet Archive]]</ref><ref>Electromagnetic theory, Volume 1. By [[奧利弗·黑維塞|Oliver Heaviside]]. "The Electrician" printing and publishing company, limited, 1893.</ref><ref>Elements of the differential calculus. By [[Wesley Stoker Barker Woolhouse]]. Weale, 1854.</ref><ref>An Elementary Treatise on the Integral Calculus: Founded on the Method of Rates Or Fluxions. By [[William Woolsey Johnson]]. John Wiley & Sons, 1881.<br />参见：[[流数法]]。</ref><ref>Vector Calculus: With Applications to Physics. By [[James Byrnie Shaw]]. D. Van Nostrand, 1922. p205.<br />参见：[[格林公式]]。</ref><ref>A Treatise on the Integral Calculus, Volume 2. By [[Joseph Edwards (Mathematician)|Joseph Edwards]]. Chelsea Publishing Company, 1922.</ref> 指出对于任意足够[[光滑函数|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。<ref>参见：
*  H. Helmholtz (1858) [http://books.google.com/books?id=6gwPAAAAIAAJ&pg=PA25#v=onepage&q&f=false "Über Integrale der hydrodynamischen Gleichungen, welcher der Wirbelbewegungen entsprechen"] {{Wayback|url=http://books.google.com/books?id=6gwPAAAAIAAJ&pg=PA25#v=onepage&q&f=false |date=20140210091227 }} (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) [http://books.google.com/books?id=L_NYAAAAYAAJ&pg=PA1#v=onepage&q&f=false "On the dynamical theory of diffraction,"] ''Transactions of the Cambridge Philosophical Society'', vol. 9, part I, pages 1-62; see pages 9-10.</ref>

这意味着任何矢量场 {{math|'''F'''}}，都可以视为两个势场（[[純量勢]] {{mvar|φ}} 和[[向量勢]] {{math|'''A'''}}）之和。

==定理內容==
假定 {{math|'''F'''}} 為定義在有界區域 {{math|''V'' ⊆ '''R'''<sup>3</sup>}} 裡的二次連續可微向量場，且 {{mvar|S}} 為 {{mvar|V}} 的包圍面，則 {{math|'''F'''}} 可被分解成無[[旋度]]及無[[散度]]兩部份：<ref>{{cite web|url=http://www.cems.uvm.edu/~oughstun/LectureNotes141/Topic_03_(Helmholtz'%20Theorem).pdf|title=Helmholtz' Theorem|publisher=University of Vermont|deadurl=yes|archiveurl=https://web.archive.org/web/20120813005804/http://www.cems.uvm.edu/~oughstun/LectureNotes141/Topic_03_(Helmholtz'%20Theorem).pdf|archivedate=2012-08-13|accessdate=2014-08-14}}</ref>
:<math>\mathbf{F}=-\boldsymbol{\nabla}\Phi+\boldsymbol{\nabla}\times\mathbf{A}</math>，

其中
:<math>\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'</math>


:<math>\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'</math>


如果 {{math|''V'' {{=}} '''R'''<sup>3</sup>}}，且 {{math|'''F'''}} 在無窮遠處消失的比 <math>1/r</math> 快，則純量勢及向量勢的第二項為零，也就是說
<ref name="griffiths">David J. Griffiths, ''Introduction to Electrodynamics'', Prentice-Hall, 1999, p. 556.</ref>

:<math>\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'</math>


:<math>\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'</math>

==推導==
假定我們有一個向量函數<math>\mathbf{F}\left(\mathbf{r}\right)</math>，且其旋度<math>\boldsymbol{\nabla}\times\mathbf{F}</math>及散度<math>\boldsymbol{\nabla}\cdot\mathbf{F}</math>已知。利用[[狄拉克δ函数]]可將函數改寫成

:<math>\delta\left(\mathbf{r}-\mathbf{r}'\right)=-\frac{1}{4\pi}\nabla^{2}\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}</math>，

:<math>\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'</math>。

利用以下等式

:<math>\nabla^{2}\mathbf{a}=\boldsymbol{\nabla}\left(\boldsymbol{\nabla}\cdot\mathbf{a}\right)-\boldsymbol{\nabla}\times\left(\boldsymbol{\nabla}\times\mathbf{a}\right)</math>，

可得

:<math>\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]</math>

:::<math>=-\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]</math>。

注意到<math>\boldsymbol{\nabla}\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}=-\boldsymbol{\nabla}'\frac{1}{\left|\mathbf{r}-\mathbf{r}'\right|}</math>，我們可將上式改寫成

:<math>\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]</math>。


利用以下二等式，

:<math>\mathbf{a}\cdot\boldsymbol{\nabla}\psi=-\psi\left(\boldsymbol{\nabla}\cdot\mathbf{a}\right)+\boldsymbol{\nabla}\cdot\left(\psi\mathbf{a}\right)</math>
:<math>\mathbf{a}\times\boldsymbol{\nabla}\psi=\psi\left(\boldsymbol{\nabla}\times\mathbf{a}\right)-\boldsymbol{\nabla}\times\left(\psi\mathbf{a}\right)</math>。

可得

:<math>\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]</math>。

利用[[散度定理]]，方程式可改寫成

:<math>\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]</math>

:::<math>=
-\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]
</math>。

定義

:<math>\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'</math>

:<math>\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'</math>

所以

:<math>\mathbf{F}=-\boldsymbol{\nabla}\Phi+\boldsymbol{\nabla}\times\mathbf{A}</math>

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

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：
:<math>\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} 
</math>
所以
:<math> \vec{\mathbf{G}}(\vec{\omega}) = - i \,\vec{\omega} \, G_\Phi(\vec{\omega})  + i \, \vec{\omega} \times \vec{\mathbf{G}}_\mathbf{A}(\vec{\omega}) </math>

:<math>
\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}
</math>

==注释==
{{reflist}}

== 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.&nbsp;92–93
* George B. Arfken and Hans J. Weber, Mathematical Methods for Physicists International Edition, 6th edition, Academic Press: San Diego (2005) pp.&nbsp;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 ==
*[http://mathworld.wolfram.com/HelmholtzsTheorem.html Helmholtz theorem] {{Wayback|url=http://mathworld.wolfram.com/HelmholtzsTheorem.html |date=20191224172544 }}（[[MathWorld]]）

{{基本定理}}

[[Category:向量分析]]
[[Category:分析定理]]
[[Category:解析几何]]
[[Category:微積分定理]]