定理內容 编辑 推導 编辑
假定我們有一個向量函數 F ( r ) {\displaystyle \mathbf {F} \left(\mathbf {r} \right)} ∇ × F {\displaystyle {\boldsymbol {\nabla }}\times \mathbf {F} } ∇ ⋅ F {\displaystyle {\boldsymbol {\nabla }}\cdot \mathbf {F} } 狄拉克δ函数 可將函數改寫成
δ ( r − r ′ ) = − 1 4 π ∇ 2 1 | r − r ′ | {\displaystyle \delta \left(\mathbf {r} -\mathbf {r} '\right)=-{\frac {1}{4\pi }}\nabla ^{2}{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}} F ( r ) = ∫ V F ( r ′ ) δ ( r − r ′ ) d V ′ = ∫ V F ( r ′ ) ( − 1 4 π ∇ 2 1 | r − r ′ | ) d V ′ = − 1 4 π ∇ 2 ∫ V F ( r ′ ) | r − r ′ | d V ′ {\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
∇ 2 a = ∇ ( ∇ ⋅ a ) − ∇ × ( ∇ × a ) {\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
F ( r ) = − 1 4 π [ ∇ ( ∇ ⋅ ∫ V F ( r ′ ) | r − r ′ | d V ′ ) − ∇ × ( ∇ × ∫ V F ( r ′ ) | r − r ′ | d V ′ ) ] {\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]} = − 1 4 π [ ∇ ( ∫ V F ( r ′ ) ⋅ ∇ 1 | r − r ′ | d V ′ ) + ∇ × ( ∫ V F ( r ′ ) × ∇ 1 | r − r ′ | d V ′ ) ] {\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]} 注意到 ∇ 1 | r − r ′ | = − ∇ ′ 1 | r − r ′ | {\displaystyle {\boldsymbol {\nabla }}{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}=-{\boldsymbol {\nabla }}'{\frac {1}{\left|\mathbf {r} -\mathbf {r} '\right|}}}
F ( r ) = − 1 4 π [ − ∇ ( ∫ V F ( r ′ ) ⋅ ∇ ′ 1 | r − r ′ | d V ′ ) − ∇ × ( ∫ V F ( r ′ ) × ∇ ′ 1 | r − r ′ | d V ′ ) ] {\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]}
a ⋅ ∇ ψ = − ψ ( ∇ ⋅ a ) + ∇ ⋅ ( ψ a ) {\displaystyle \mathbf {a} \cdot {\boldsymbol {\nabla }}\psi =-\psi \left({\boldsymbol {\nabla }}\cdot \mathbf {a} \right)+{\boldsymbol {\nabla }}\cdot \left(\psi \mathbf {a} \right)} a × ∇ ψ = ψ ( ∇ × a ) − ∇ × ( ψ a ) {\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
F ( r ) = − 1 4 π [ − ∇ ( − ∫ V ∇ ′ ⋅ F ( r ′ ) | r − r ′ | d V ′ + ∫ V ∇ ′ ⋅ F ( r ′ ) | r − r ′ | d V ′ ) − ∇ × ( ∫ V ∇ ′ × F ( r ′ ) | r − r ′ | d V ′ − ∫ V ∇ ′ × F ( r ′ ) | r − r ′ | d V ′ ) ] {\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]} 利用散度定理 ，方程式可改寫成
F ( r ) = − 1 4 π [ − ∇ ( − ∫ V ∇ ′ ⋅ F ( r ′ ) | r − r ′ | d V ′ + ∮ S n ^ ′ ⋅ F ( r ′ ) | r − r ′ | d S ′ ) − ∇ × ( ∫ V ∇ ′ × F ( r ′ ) | r − r ′ | d V ′ − ∮ S n ^ ′ × F ( r ′ ) | r − r ′ | d S ′ ) ] {\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]} = − ∇ [ 1 4 π ∫ V ∇ ′ ⋅ F ( r ′ ) | r − r ′ | d V ′ − 1 4 π ∮ S n ^ ′ ⋅ F ( r ′ ) | r − r ′ | d S ′ ] + ∇ × [ 1 4 π ∫ V ∇ ′ × F ( r ′ ) | r − r ′ | d V ′ − 1 4 π ∮ S n ^ ′ × F ( r ′ ) | r − r ′ | d S ′ ] {\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]} 定義
Φ ( r ) ≡ 1 4 π ∫ V ∇ ′ ⋅ F ( r ′ ) | r − r ′ | d V ′ − 1 4 π ∮ S n ^ ′ ⋅ F ( r ′ ) | r − r ′ | d S ′ {\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'} A ( r ) ≡ 1 4 π ∫ V ∇ ′ × F ( r ′ ) | r − r ′ | d V ′ − 1 4 π ∮ S n ^ ′ × F ( r ′ ) | r − r ′ | 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
F = − ∇ Φ + ∇ × A {\displaystyle \mathbf {F} =-{\boldsymbol {\nabla }}\Phi +{\boldsymbol {\nabla }}\times \mathbf {A} } 利用傅利葉轉換做推導 编辑 （疑似有错误）將F 改寫成傅利葉轉換 的形式：
F → ( r → ) = ∭ G → ( ω → ) e i ω → ⋅ r → d ω → {\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：
G Φ ( ω → ) = i G → ( ω → ) ⋅ ω → | | ω → | | 2 G → A ( ω → ) = i ω → × ( G → ( ω → ) + i G Φ ( ω → ) ω → ) Φ ( r → ) = ∭ G Φ ( ω → ) e i ω → ⋅ r → d ω → A → ( r → ) = ∭ G → A ( ω → ) e i ω → ⋅ r → d ω → {\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
G → ( ω → ) = − i ω → G Φ ( ω → ) + i ω → × G → A ( ω → ) {\displaystyle {\vec {\mathbf {G} }}({\vec {\omega }})=-i\,{\vec {\omega }}\,G_{\Phi }({\vec {\omega }})+i\,{\vec {\omega }}\times {\vec {\mathbf {G} }}_{\mathbf {A} }({\vec {\omega }})} F → ( r → ) = − ∭ i ω → G Φ ( ω → ) e i ω → ⋅ r → d ω → + ∭ i ω → × G → A ( ω → ) e i ω → ⋅ r → d ω → = − ∇ Φ ( r → ) + ∇ × A → ( r → ) {\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 编辑
^ On Helmholtz's Theorem in Finite Regions. By Jean Bladel. Midwestern Universities Research Association, 1958. ^ Hermann von Helmholtz. Clarendon Press, 1906. By Leo Koenigsberger. p357 ^ An Elementary Course in the Integral Calculus. By Daniel Alexander Murray. American Book Company, 1898. p8. ^ J. W. Gibbs & Edwin Bidwell Wilson (1901) Vector Analysis , page 237, link from Internet Archive ^ Electromagnetic theory, Volume 1. By Oliver Heaviside . "The Electrician" printing and publishing company, limited, 1893. ^ Elements of the differential calculus. By Wesley Stoker Barker Woolhouse. Weale, 1854. ^ An Elementary Treatise on the Integral Calculus: Founded on the Method of Rates Or Fluxions. By William Woolsey Johnson. John Wiley & Sons, 1881.流数法 。 ^ Vector Calculus: With Applications to Physics. By James Byrnie Shaw. D. Van Nostrand, 1922. p205.格林公式 。 ^ A Treatise on the Integral Calculus, Volume 2. By Joseph Edwards. Chelsea Publishing Company, 1922. ^ 参见：H. Helmholtz (1858) "Über Integrale der hydrodynamischen Gleichungen, welcher der Wirbelbewegungen entsprechen" （页面存档备份 ，存于互联网档案馆 ） (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. ^ Helmholtz' Theorem (PDF) . University of Vermont. [2014-08-14 ] . （原始内容 (PDF) 存档于2012-08-13）. ^ David J. Griffiths, Introduction to Electrodynamics , Prentice-Hall, 1999, p. 556. 参考文献 编辑
一般参考文献 编辑 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 编辑 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 编辑 ，
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![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8762db3c85abbb95c1bb3d144af529c16850cd90)
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![{\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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a14c3123be2a9c934f5aa4018d54503065b0c46)
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