Volume form

In mathematics, the volume form is a differential form that represents a unit volume of a Riemannian manifold or a pseudo-Riemannian manifold. In local coordinates, it can be expressed as

<math>\omega = \sqrt{|g|} dx^1\wedge ... \wedge dx^n<math>

where the manifold is an n dimensional manifold. Here, <math>|g|<math> is the absolute value of the determinant of the metric tensor on the manifold. The <math>dx^i<math> are the 1-forms providing a basis for the cotangent bundle of the manifold.

A number of different notations are in use for the volume form. These include

<math>\omega = \mathrm{vol}_n = \epsilon = *(1)<math>

Here, the * is the Hodge dual, thus the last form, *(1), emphasizes that the volume form is the Hodge dual of the trival constant map on the manifold.

Although the greek letter ω is frequently used to denote teh volume form, this notation is hardly universal; the symbol ω often carries many other meanings in differential geometry; thus, the appearence of ω in a formula does not necessarily mean that it is the volume form.