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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Helmholtz decomposition ： ウィキペディア英語版 Helmholtz decomposition In physics and mathematics, in the area of vector calculus, Helmholtz's theorem,〔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〕 also known as the fundamental theorem of vector calculus,〔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. See also: Method of Fluxions.〕〔Vector Calculus: With Applications to Physics. By James Byrnie Shaw. D. Van Nostrand, 1922. p205. See also: Green's Theorem.〕〔A Treatise on the Integral Calculus, Volume 2. By Joseph Edwards. Chelsea Publishing Company, 1922.〕 states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition. It is named after Hermann von Helmholtz.〔See: * 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.〕 Because an irrotational vector field has a scalar potential and a solenoidal vector field has a vector potential, the Helmholtz decomposition states that a vector field (satisfying appropriate smoothness and decay conditions) can be decomposed as the sum of the form $-\backslash operatorname\; \backslash Phi\; +\; \backslash operatorname\backslash mathbf$ where is a scalar field, called scalar potential, and is a vector field called a vector potential. ==Statement of the theorem== Let be a vector field on a bounded domain , which is twice continuously differentiable, and let be the surface that encloses the domain . Then can be decomposed into a curl-free component and a divergence-free component:〔(【引用サイトリンク】title=Helmholtz' Theorem )〕 :$\backslash mathbf=-\backslash nabla\backslash Phi+\backslash nabla\backslash times\backslash mathbf,$ where :$\backslash Phi(\backslash mathbf)=\backslash frac\backslash int\_\backslash frac\text{'}\backslash right)\}\text{'}\backslash right|\}\backslash mathrmV\text{'}\; -\backslash frac\; \backslash oint\_\; \backslash mathbf\backslash left(\backslash mathbf\text{'}\backslash right)\}\text{'}\backslash right|\}\backslash mathrmS\text{'}$ :$\backslash mathbf(\backslash mathbf)=\backslash frac\backslash int\_\backslash frac\text{'}\backslash right)\}\text{'}\backslash right|\}\backslash mathrmV\text{'}\; -\backslash frac\backslash oint\_\backslash mathbf\backslash left(\backslash mathbf\text{'}\backslash right)\}\text{'}\backslash right|\}\backslash mathrmS\text{'}$ If and is therefore unbounded, and vanishes faster than $1/r$ as , then the second component of both scalar and vector potential are zero. That is,〔David J. Griffiths, ''Introduction to Electrodynamics'', Prentice-Hall, 1999, p. 556.〕 :$\backslash Phi(\backslash mathbf)=\backslash frac\backslash int\_\backslash left(\backslash mathbf\text{'}\backslash right)\}\text{'}\backslash right|\}\backslash mathrmV\text{'}$ :$\backslash mathbf\; (\backslash mathbf)=\backslash frac\backslash int\_\backslash left(\backslash mathbf\text{'}\backslash right)\}\text{'}\backslash right|\}\backslash mathrmV\text{'}$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Helmholtz decomposition」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.