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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 Prolate spheroidal coordinates ： ウィキペディア英語版 Prolate spheroidal coordinates Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates. Prolate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two smallest principal axes are equal in length. Prolate spheroidal coordinates can be used to solve various partial differential equations in which the boundary conditions match its symmetry and shape, such as solving for a field produced by two centers, which are taken as the foci on the ''z''-axis. One example is solving for the wavefunction of an electron moving in the electromagnetic field of two positively charged nuclei, as in the hydrogen molecular ion, H2+. Another example is solving for the electric field generated by two small electrode tips. Other limiting cases include areas generated by a line segment (μ=0) or a line with a missing segment (ν=0). ==Definition== The most common definition of prolate spheroidal coordinates $(\backslash mu,\; \backslash nu,\; \backslash phi)$ is :$$ x = a \ \sinh \mu \ \sin \nu \ \cos \phi :$$ y = a \ \sinh \mu \ \sin \nu \ \sin \phi :$$ z = a \ \cosh \mu \ \cos \nu where $\backslash mu$ is a nonnegative real number and $\backslash nu\; \backslash in\; (\backslash pi)$. The azimuthal angle $\backslash phi$ belongs to the interval $(2\backslash pi)$. The trigonometric identity :$$ \frac \cosh^ \mu} + \frac} \mu} = \cos^ \nu + \sin^ \nu = 1 shows that surfaces of constant $\backslash mu$ form prolate spheroids, since they are ellipses rotated about the axis joining their foci. Similarly, the hyperbolic trigonometric identity :$$ \frac \cos^ \nu} - \frac} \nu} = \cosh^ \mu - \sinh^ \mu = 1 shows that surfaces of constant $\backslash nu$ form hyperboloids of revolution. 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「Prolate spheroidal coordinates」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.