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volume element
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(Definition)
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If  is an  dimensional manifold, then a differential  form that is never zero is called a volume element or a volume form. Usually one volume form is associated with the manifold. The volume element is sometimes denoted by   or
 If the manifold is a Riemannian manifold with metric  then the natural volume form is defined in local coordinates
 by
It is not hard to show that a manifold has a volume form if and only if it is orientable.
If the manifold is
 then the usual volume element
 is called the Euclidean volume element or Euclidean volume form. In this context,
 is usually treated as
 unless stated otherwise.
When  , then the form is frequently called the area element or area form and frequently denoted by  . Furthermore, when the manifold is a submanifold of
 , then many authors will refer to a surface area element or surface area form.
When the context is measure theoretic, this form is sometimes called a volume measure, area measure, etc...
- 1
- Michael Spivak. Calculus on Manifolds, W.A. Benjamin, Inc., 1965.
- 2
- William M. Boothby. An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic Press, San Diego, California, 2003.
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"volume element" is owned by jirka.
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(view preamble)
| Other names: |
volume form, volume measure |
| Also defines: |
area element, area form, area measure, Euclidean volume element, Euclidean volume form, euclidean volume measure, surface area measure, surface area element, surface area form |
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Cross-references: measure, submanifold, orientable, local coordinates, Riemannian manifold, manifold
There are 17 references to this entry.
This is version 2 of volume element, born on 2007-12-12, modified 2007-12-12.
Object id is 10122, canonical name is VolumeElement.
Accessed 1565 times total.
Classification:
| AMS MSC: | 53-00 (Differential geometry :: General reference works ) | | | 58A10 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Differential forms) |
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Pending Errata and Addenda
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