Differential equations | Differential geometry
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 f2(dx2 + dy2) on a surface of constant Gaussian curvature K: where ∆0 is the flat Laplace operator Liouville's equation appears in the study of isothermal coordinates in differential geometry: the independent variables x,y are the coordinates, while f can be described as the conformal factor with respect to the flat metric. Occasionally it is the square f2 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. (Wikipedia).
Summary for graph an equation in Standard form
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