Talk:Volume form

What a ludicrously mis-pitched article

Would someone capable of understanding this page really be looking up "volume element" on wikipedia? - 68.101.67.210

Relevance of this question? :) Marc 02:52, 11 May 2007 (UTC)

To my mind, someone looking for "volume element" is coming looking for examples of volume elements in different co-ordinate systems rather than for an incomprehensible introduction to the world of line bundles. There's nothing wrong with having a more formal "introduction" on the page, but I'd suggest that there should be something that you don't need a degree in pure maths to understand, too. —Preceding unsigned comment added by 88.66.11.4 (talk) 09:10, 14 October 2008 (UTC)

I agree in part that the article should try to say at least something moderately accessible. I don't believe that this should be the chief aim of the article, however. siℓℓy rabbit (talk) 06:23, 15 October 2008 (UTC)

Odd/twisted forms

It would be nice to specify the difference between even n-forms and odd or twisted n-forms. The latter, also called volume densities, permit to define "volume elements" also on non-orientable manifolds. See the various books and articles by Burke, Bressoud, the book by Marsden & Abraham & Ratiu, and the book by Choquet-Bruhat et al. I shall possibly write some more on this. —Preceding unsigned comment added by 83.233.183.58 (talk) 10:47, 9 November 2007 (UTC)

No local structure

Are there any references you could suggest for the "no local structure" theorems mentioned? I've looked at the original work by Moser, but I'd appreciate any other suggestions. —Preceding unsigned comment added by 130.207.104.54 (talk) 19:32, 4 November 2008 (UTC)

determinant of a metric tensor =

Hello, what is the determinant of a metric tensor? Failed to find this here. —Preceding unsigned comment added by 128.176.163.95 (talk) 16:14, 24 October 2008 (UTC)

There is a link to the metric tensor page which explains that the metric determines a bilinear form at each point on the manifold which varies smoothly from point to point. At any point the bilinear form can be represented by a matrix (depending on the chosen coordinates). The determinant of the metric tensor is the determinant of this matrix. —Preceding unsigned comment added by 195.112.46.6 (talk) 00:06, 30 January 2009 (UTC)

Skew-symmetric

Shouldn't volume forms be skew-symmetric? The introduction doesn't seem to say so. ~~ Dr Dec (Talk) ~~ 20:20, 20 October 2009 (UTC)

This is part of the definition of a differential form. 173.75.156.204 (talk) 20:51, 20 October 2009 (UTC)

Poorly written

The section Riemannian volume form begins as follows:

"Any oriented pseudo-Riemannian (including Riemannian) manifold has a natural volume form. In local coordinates, it can be expressed as $$\omega ={\sqrt {|g|}}dx^{1}\wedge \dots \wedge dx^{n}$$ where the $dx^{i}$ are 1-forms that form a positively oriented basis for the cotangent bundle of the manifold. Here, $|g|$ is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold."

This paragraph is screaming to the reader: "If you don't understand this, that is just too bad for you, because I am not going to explain what I am talking about. (Like far, far too many mathematics articles in Wikipedia.)

The main problem is the final sentence, "Here, $|g|$ is the absolute value of the determinant of the matrix representation of the metric tensor on the manifold."

This never states how this "matrix representation" is supposed to be obtained. Or, is any matrix representation just as good as any other here???

I happen to be trained in Riemannian manifolds, so I know the answer. But the person who wrote this paragraph is evidently also trained in Riemannian manifolds, so that person ought to finish what they began and write this so that a beginner would know what is intended.

It doesn't matter what the excuse is: This is poorly written no matter how one views it.