# Liouville's 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 f2(dx2 + dy2) on a surface of constant Gaussian curvature K:

$\Delta _{0}\log f=-Kf^{2},$ For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).
For Liouville's equation in quantum mechanics, see Von Neumann equation.
For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelfand equation.

where 0 is the flat Laplace operator

$\Delta _{0}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}=4{\frac {\partial }{\partial z}}{\frac {\partial }{\partial {\bar {z}}}}.$ 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.