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Liouville 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(\mathrmx^2 + \mathrmy^2) on a surface of constant Gaussian curvature :
:\Delta_0\log f = -K f^2,
where \Delta_0 is the flat Laplace operator.
:\Delta_0 = \frac +\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 "\log \,f = u", another commonly found form of Liouville's equation is obtained:
:\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\log \,f = u" of the previous change of variables and Wirtinger calculus:〔See .〕
:\Delta_0 u = - 2K e^\quad\Longleftrightarrow\quad \frac = - \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:
:\frac + \frac = e^f


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