# Liouville's equation

In differential geometry, **Liouville's equation**, named after Joseph Liouville,[1][2] is the nonlinear partial differential equation satisfied by the conformal factor f of a metric *f*^{2}(d*x*^{2} + d*y*^{2}) on a surface of constant Gaussian curvature K:

*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

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 *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.[3]