Volume form

A volume shape is a mathematical object that is required for integration over spatial regions, in particular the use of specific coordinate systems, which is a special case of a volume.

In physics and engineering also names as infinitesimal volume element or scaling factor are used.

Examples in 3 dimensions

• Cartesian coordinates:
• Cylindrical coordinates:
• Spherical coordinates:

Mathematical definition

From a mathematical point of view a volume form on one -dimensional manifold is a differential form of degree. In the case of an oriented Riemannian manifold there is a canonical volume form of the metric used, which takes the value 1 on a positively oriented orthonormal basis. This is called the Riemannian volume form.

Integration with volume forms

Is a volume shape on a manifold, and a function can be integrated, then the integral

Defined above local maps as follows: It may be local coordinates so that

Is positively oriented. Then you can map the area as

Write; the integral is then the ordinary Lebesgue integral of. For the integral over the whole of a partition of unity or a decomposition of the manifold into disjoint measurable subsets can be used. From the transformation theorem shows that this definition is independent of cards.