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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 Poloidal–toroidal decomposition ： ウィキペディア英語版 Poloidal–toroidal decomposition In vector calculus, a poloidal–toroidal decomposition is a restricted form of the Helmholtz decomposition that is often used in the spherical-coordinates analysis of solenoidal vector fields, for example, magnetic fields and incompressible fluids. For a three-dimensional F, such that :$\backslash nabla\; \backslash cdot\; \backslash mathbf\; =\; 0,$ can be expressed as the sum of a toroidal and poloidal vector fields: :$\backslash mathbf\; =\; \backslash mathbf\; +\; \backslash mathbf\; =\; \backslash nabla\; \backslash times\; \backslash Psi\; \backslash mathbf\; +\; \backslash nabla\; \backslash times\; (\backslash nabla\; \backslash times\; \backslash Phi\; \backslash mathbf),$ where $\backslash mathbf$ is a radial vector in spherical coordinates $(r,\backslash theta,\backslash phi)$, and where $\backslash mathbf$ is a toroidal field :$\backslash mathbf\; =\; \backslash nabla\; \backslash times\; \backslash Psi\; \backslash mathbf$ for scalar field $\backslash Psi\; (r,\backslash theta,\backslash phi)$, and where $\backslash mathbf$ is a poloidal field :$\backslash mathbf\; =\; \backslash nabla\; \backslash times\; \backslash nabla\; \backslash times\; \backslash Phi\; \backslash mathbf$ for scalar field $\backslash Phi\; (r,\backslash theta,\backslash phi)$. This decomposition is symmetric in that the curl of a toroidal field is poloidal, and the curl of a poloidal field is toroidal. A toroidal vector field is tangential to spheres around the origin :$\backslash mathbf\; \backslash cdot\; \backslash mathbf\; =\; 0$, while the curl of a poloidal field is tangential to those spheres :$\backslash mathbf\; \backslash cdot\; (\backslash nabla\; \backslash times\; \backslash mathbf)\; =\; 0$. The poloidal–toroidal decomposition is unique if it is required that the average of the scalar fields $\backslash Psi$ and $\backslash Phi$ vanishes on every sphere of radius $r$. == Cartesian decomposition == A poloidal–toroidal decompositions also exist in Cartesian coordinates, but a mean-field flow has to included in this case. For example, every solenoidal vector field can be written as : $\backslash mathbf(x,y,z)\; =\; \backslash nabla\; \backslash times\; g(x,y,z)\; \backslash hat\})\; +\; b\_x(z)\; \backslash hat\},$ where $\backslash hat\},\; \backslash hat\{\backslash mathbf\{z\}\}$ denote the unit vectors in the coordinate directions. 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「Poloidal–toroidal 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.