亥姆霍兹分解

假定 F 為定義在有界區域 V ⊆ R³ 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：

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

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 = R³，且 F 在無窮遠處消失的比 ${\displaystyle 1/r}$ 快，則純量勢及向量勢的第二項為零，也就是說

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

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

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

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

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

利用以下等式

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

可得

${\displaystyle \mathbf {F} \left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\left}$

${\displaystyle =-{\frac {1}{4\pi }}\left}$。

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

${\displaystyle \mathbf {F} \left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\left}$。

利用以下二等式，

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

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

可得

${\displaystyle \mathbf {F} \left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\left}$。

利用散度定理，方程式可改寫成

${\displaystyle \mathbf {F} \left(\mathbf {r} \right)=-{\frac {1}{4\pi }}\left}$

${\displaystyle =-{\boldsymbol {\nabla }}\left+{\boldsymbol {\nabla }}\times \left}$。

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

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

（疑似有错误） 將F改寫成傅利葉轉換的形式：

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

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：

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

所以

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

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

• 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

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• Helmholtz theorem （页面存档备份，存于）（MathWorld）