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: ''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 of a metric on a surface of constant Gaussian curvature : : where is the flat Laplace operator. : 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 that is referred to as the conformal factor, instead of 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 "", another commonly found form of Liouville's equation is obtained: : Other two forms of the equation, commonly found in the literature,〔See and .〕 are obtained by using the slight variant "" of the previous change of variables and Wirtinger calculus:〔See .〕 : 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: : 〕 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
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