在物理学和数学中的向量分析中，**亥姆霍兹定理**， 或称**向量分析基本定理**， 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。

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* ⊆ **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：

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

\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'

  

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

\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

{\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

{\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]。
定義

\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'
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：

{\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

{\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}}

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 （页面存档备份，存于）（MathWorld）

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