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:
- 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 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.