Talk:Liouville's equation

Disambiguate Liouville

It would be nice to create a disambiguation page for Liouville equation. Currently the similarly-named equation from physics hogs up all the possible variations of the name. Katzmik 14:11, 8 August 2007 (UTC)

Is it possible to merge the two articles? The main one already seems to have a section on geometry. In any case, it would be better to call the article Louiville's equation (differential geometry) than to give it the wrong name just because the better ones are "already taken." This is an encyclopedia, not a domain name registrar.. ;-) -- SCZenz 14:15, 8 August 2007 (UTC)

Merging the two articles would not seem to be appropriate, as the inequalities are unrelated as far as I can see (I have seen some applications of the curvature equation in physics, and perhaps ultimately they are related, so I should say that I don't see the connection). Actually wiki is a kind of a "domain name registrar" since someone looking for information on Liouville's equation will not think of typing "(differential geometry)" in the "search" window. I think this case is actually very similar to the Wirtinger inequality, where the disambiguation page was created. Katzmik 07:21, 9 August 2007 (UTC)

If you think it's best, by all means go ahead. The only thing you might need help with is deleting pages to make room for moves—let me know if so. -- SCZenz 07:32, 9 August 2007 (UTC)

You could also put a note in italics at the top of the physics article--see the top of Cosmic variance for an example--and then link to Liouville equations (differential geometry) or similar. That's the usual way to do it when one use of a term is very common, but people might be looking for another. -- SCZenz 07:34, 9 August 2007 (UTC)

Again, referring to the physics term as being very common is true with reference to physics articles. In differential geometry, the curvature equation is more common. Katzmik 08:33, 9 August 2007 (UTC)

Liouville's equation is a partial differential equation

As I remarked in edit, Liouville's equation is not an ordinary differential equation, but it is a partial differential equation, since the associated differential operator and therefore its solution is a function of two independent variables: also, having a look to its general solution, it can bee seen that it depends on a function, not on a finite number of parameters. Daniele.tampieri (talk) 12:21, 1 January 2014 (UTC)

Misleading introduction?

The article begins as follows:

"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:

$\Delta _{0}\log f=-Kf^{2},$

where ∆₀ is the flat Laplace operator"

This is not false!

But typically people think of the curvature as secondary to the riemannian metric, which is primary (OK, after the diffeomorphism class of the manifold it is defined on.)

So, normally (these days!) the curvature K would be expressed in terms of the laplacian of the logarithm of f(z) (where f(z) > 0 for all z), for the conformal metric f(z)² (dx²+dy²). In other words, the currently displayed equation would be expressed instead as a formula for K. )

But, perhaps the way it is described in the article as the solution to a certain PDE is how Liouville described his theorem? If so, this is most definitely worth mentioning explicitly in the article.