Liouville's equation

: "For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian)."In differential geometry, Liouville's equation, named after Joseph Liouville, is the equation satisfied by the conformal factor "f" of a metric $f^2\; (dx^2\; +\; dy^2)$ on a surface of constant Gaussian curvature "K":

:$Delta\_0\; ;log\; f\; =\; -K\; f^2,$

where $Delta\_0$ is the flat Laplace operator.

:$Delta\_0\; =\; frac\{partial^2\}\{partial\; x^2\}\; +frac\{partial^2\}\{partial\; y^2\}$

Liouville's equation typically appears in differential geometry books under the heading isothermal coordinates. This term refers to the coordinates "x,y", while "f" can be described as the conformal factor with respect to the flat metric (sometimes the square $f^2$ is referred to as the conformal factor, instead of "f" itself).

Replacing "f" by $u=log\; ,f$, we obtain another commonly found form of the same equation:

:$Delta\_0\; u\; =\; -\; K\; e^\{2u\}.$

Laplace-Beltrami operator

In a more invariant fashion, the equation can be written in terms of the "intrinsic" Laplace-Beltrami operator

: $Delta\_\{mathrm\{LB\; =\; frac\{1\}\{f^2\}\; Delta\_0$

as follows:

:$Delta\_\{mathrm\{LBlog;\; f\; =\; -K.$