Volume form

volume element.

Let be a vector space of dimension with a given orientation and an inner product. The corresponding volume form, or volume element, is the unique element , the space of -forms on (cf. Exterior form), such that for each orthonormal (with respect to the given inner product) basis of . Recall that is one-dimensional. If with the standard inner product and orientation, then for each -tuple of vectors (written out in the standard basis for calculating the determinant) and is the volume of the parallelopipedon spanned by the line segments from zero to the .

If is an oriented Riemannian manifold, then the volume form on is defined by requiring that for each is the unique volume element on defined by inner product and orientation on each . One often writes for the volume form on , even though there may not be an -form on of which it is the exterior derivative.

In given local coordinates , let be the two-form (matrix) determining the inner product on (with respect to the basis , cf. Tangent vector). Then in local coordinates,

where depending on whether the orientation of corresponds to the standard one on or not (under the given coordinate chart).

On a Riemannian manifold a function is integrated by intergrating the -form over in the sense of integration on manifolds.

Let denote the Hodge star operator (cf. Laplace operator). The divergence of a vector field, locally given by , is defined as the function

One then has

and, on integration over an -chain in , by the Stokes formula there results the higher-dimensional divergence theorem, which specializes to the usual one for a -dimensional submanifold with boundary in .

References

[a1]	M. Spivak, "Calculus on manifolds" , Benjamin (1965)
[a2]	M. Hazewinkel, "A tutorial introduction to differentiable manifolds and calculus on manifolds" W. Schiehlen (ed.) W. Wedig (ed.) , Analysis and estimation of stochastic mechanical systems , Springer (Wien) (1988) pp. 316–340
[a3]	Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick, "Analysis, manifolds, and physics" , North-Holland (1977) (Translated from French)