亥姆霍茲分解

物理學數學中的向量分析中,亥姆霍茲定理[1][2] 或稱向量分析基本定理[3][4][5][6][7][8][9] 指出對於任意足夠光滑、快速衰減的三維向量場可分解為一個無旋向量場和一個螺線向量場的和,這個過程被稱作亥姆霍茲分解。此定理以物理學家赫爾曼·馮·亥姆霍茲為名。[10]

這意味着任何向量場 F,都可以視為兩個勢場(純量勢 φ向量勢 A)之和。

定理內容

假定 F 為定義在有界區域 VR3 裏的二次連續可微向量場,且 SV 的包圍面,則 F 可被分解成無旋度及無散度兩部份:[11]

 

其中

 


 


如果 V = R3,且 F 在無窮遠處消失的比   快,則純量勢及向量勢的第二項為零,也就是說 [12]

 


 

推導

假定我們有一個向量函數 ,且其旋度 及散度 已知。利用狄拉克δ函數可將函數改寫成

 
 

利用以下等式

 

可得

 
 

注意到 ,我們可將上式改寫成

 


利用以下二等式,

 
 

可得

 

利用散度定理,方程式可改寫成

 
 

定義

 
 

所以

 

利用傅利葉轉換做推導

(疑似有錯誤) 將F改寫成傅利葉轉換的形式:

 

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:

 

所以

 
 

註釋

  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.
    參見:流數法
  8. ^ Vector Calculus: With Applications to Physics. By James Byrnie Shaw. D. Van Nostrand, 1922. p205.
    參見:格林公式
  9. ^ A Treatise on the Integral Calculus, Volume 2. By Joseph Edwards. Chelsea Publishing Company, 1922.
  10. ^ 參見:
  11. ^ Helmholtz' Theorem (PDF). University of Vermont. [2014-08-14]. (原始內容 (PDF)存檔於2012-08-13). 
  12. ^ David J. Griffiths, Introduction to Electrodynamics, Prentice-Hall, 1999, p. 556.

參考文獻

一般參考文獻

  • 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

弱形式的參考文獻

  • 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.

外部連結