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， 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。此定理以物理学家赫尔曼·冯·亥姆霍兹为名。

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

定理内容

假定 F 为定义在有界区域 V ⊆ R³ 里的二次连续可微向量场，且 S 为 V 的包围面，则 F 可被分解成无旋度及无散度两部份：

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

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 = R³，且 F 在无穷远处消失的比 $1/r$ 快，则纯量势及向量势的第二项为零，也就是说

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

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

推导

假定我们有一个向量函数$\mathbf {F} \left(\mathbf {r} \right)$，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函数可将函数改写成

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

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

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${\boldsymbol {\nabla }}{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}=-{\boldsymbol {\nabla }}'{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}$，我们可将上式改写成

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

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，方程式可改写成

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

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

定义

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

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

所以

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

利用傅利叶转换做推导

（疑似有错误） 将F改写成傅利叶转换的形式：

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

纯量场的傅利叶转换是一个纯量场，向量场的傅利叶转换是一个维度相同的向量场。 现在考虑以下纯量场及向量场：

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

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，存于互联网档案馆）（MathWorld）