# 亥姆霍兹分解

## 定理內容

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

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

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

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

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

## 推導

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

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

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

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

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

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

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

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

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

### 利用傅利葉轉換做推導

F改寫成傅利葉轉換的形式：

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

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

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

## 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 (PDF). 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.