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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 Liouville's equation ： ウィキペディア英語版 Liouville's equation : ''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor $f$ of a metric $f^2(\backslash mathrmx^2\; +\; \backslash mathrmy^2)$ on a surface of constant Gaussian curvature : :$\backslash Delta\_0\backslash log\; f\; =\; -K\; f^2,$ where $\backslash Delta\_0$ is the flat Laplace operator. :$\backslash Delta\_0\; =\; \backslash frac\; +\backslash frac$ = 4 \frac \frac Liouville's equation appears in the study of isothermal coordinates in differential geometry: the independent variables are the coordinates, while can be described as the conformal factor with respect to the flat metric. Occasionally it is the square $f^2$ that is referred to as the conformal factor, instead of $f$ itself. Liouville's equation was also taken as an example by David Hilbert in the formulation of his nineteenth problem.〔See : Hilbert does not cite explicitly Joseph Liouville.〕 ==Other common forms of Liouville's equation== By using the change of variables "$\backslash log\; \backslash ,f\; =\; u$", another commonly found form of Liouville's equation is obtained: :$\backslash Delta\_0\; u\; =\; -\; K\; e^.$ Other two forms of the equation, commonly found in the literature,〔See and .〕 are obtained by using the slight variant "$2\backslash log\; \backslash ,f\; =\; u$" of the previous change of variables and Wirtinger calculus:〔See .〕 :$\backslash Delta\_0\; u\; =\; -\; 2K\; e^\backslash quad\backslash Longleftrightarrow\backslash quad\; \backslash frac\; =\; -\; \backslash frac\; e^.$ Note that it is exactly in the first one of the preceding two forms that Liouville's equation was cited by David Hilbert in the formulation of his nineteenth problem.〔〔Hilbert assumes , therefore the equation appears as the following semilinear elliptic equation: :$\backslash frac\; +\; \backslash frac\; =\; e^f$ 〕 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「Liouville's 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.