If is an dimensional manifold, then a differential form (http://planetmath.org/DifferentialForms) 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 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. , W.A. Benjamin, Inc., 1965.
- 2 William M. Boothby. , Academic Press, San Diego, California, 2003.
|Date of creation||2013-03-22 17:40:58|
|Last modified on||2013-03-22 17:40:58|
|Last modified by||jirka (4157)|
|Defines||Euclidean volume element|
|Defines||Euclidean volume form|
|Defines||euclidean volume measure|
|Defines||surface area measure|
|Defines||surface area element|
|Defines||surface area form|