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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 Beltrami equation ： ウィキペディア英語版 Beltrami equation In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation :$.$ for ''w'' a complex distribution of the complex variable ''z'' in some open set ''U'', with derivatives that are locally ''L''2, and where μ is a given complex function in ''L''∞(''U'') of norm less than 1, called the Beltrami coefficient. Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric. Various techniques have been developed for solving the equation. The most powerful, developed in the 1950s, provides global solutions of the equation on C and relies on the L''p'' theory of the Beurling transform, a singular integral operator defined on L''P''(C) for all 1 < ''p'' < ∞. The same method applies equally well on the unit disk and upper half plane and plays a fundamental role in Teichmüller theory and the theory of quasiconformal mappings. Various uniformization theorems can be proved using the equation, including the measurable Riemann mapping theorem and the simultaneous uniformization theorem. The existence of conformal weldings can also be derived using the Beltrami equation. One of the simplest applications is to the Riemann mapping theorem for simply connected bounded open domains in the complex plane. When the domain has smooth boundary, elliptic regularity for the equation can be used to show that the uniformizing map from the unit disk to the domain extends to a C∞ function from the closed disk to the closure of the domain. ==Metrics on planar domains== Let ''U'' be an open set in C and let :$\backslash displaystyle$ be a smooth metric on ''U'', so that :$\backslash displaystyle\}$ is positive real matrix (''E'' > 0, ''G'' > 0, ''EG'' − ''F''2 > 0) varying smoothly. The Beltrami coefficient of this metric is defined by :$\backslash displaystyle$ The coefficient has modulus strictly less than one since As usual complex partial derivatives are defined by :$\backslash displaystyle\}=\; (\backslash partial\_x\; +i\backslash partial\_y).\}$ If ''f''(''x'',''y'') =(''u''(''x'',''y''),''v''(''x'',''y'')) is a smooth diffeomorphism of ''U'' onto another open set in ''V'' preserving the orientation then the standard Euclidean metric ''ds''2 = ''dx''2 + ''dy''2 induces a metric on ''U'' given by :$\backslash displaystyle$ Thus :$\backslash displaystyle|^2=\; |f\_z|^2\; |dz\; +\; d\backslash overline|^2.\}$ The Beltrami coefficient of the induced metric is given by the formula :$\backslash displaystyle.\}$ In this case :$\backslash displaystyle$ since the Jacobian of ''f'' is positive. Conversely given a metric on ''U'' a coordinate change on ''U'' making the metric conformally equivalent to the metric is a diffeomorphism ''h'' of ''U'' such that the original metric is a positive smooth function times the induced Euclidean metric; the new coordinates are called isothermal coordinates. From the formulas above this occurs if :$\backslash displaystyle,\}$ where μ is the Beltrami coefficient of the original metric. Thus isothermal coordinates can be determined locally using solutions of the Beltrami equation. 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「Beltrami equation」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.