# volume element

If $M$ is an $n$ dimensional manifold, then a differential $n$ 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 $dV,$ $\omega$ or $\operatorname{vol}_{n}.$ If the manifold is a Riemannian manifold with $g,$ then the natural volume form is defined in local coordinates $x^{1}\ldots x^{n}$ by

 $dV:=\sqrt{\lvert g\rvert}dx^{1}~{}\wedge\ldots\wedge~{}dx^{n}.$

It is not hard to show that a manifold has a volume form if and only if it is orientable.

If the manifold is ${\mathbb{R}}^{n},$ then the usual volume element $dV=dx^{1}~{}\wedge~{}dx^{2}~{}\wedge\ldots\wedge~{}dx^{n}$ is called the Euclidean volume element or Euclidean volume form. In this context, ${\mathbb{C}}^{n}$ is usually treated as ${\mathbb{R}}^{2n}$ unless stated otherwise.

When $n=2$, then the form is frequently called the area element or area form and frequently denoted by $dA$. Furthermore, when the manifold is a submanifold of ${\mathbb{R}}^{3}$, 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…

## References

• 1 Michael Spivak. , W.A. Benjamin, Inc., 1965.
• 2 William M. Boothby. , Academic Press, San Diego, California, 2003.
 Title volume element Canonical name VolumeElement Date of creation 2013-03-22 17:40:58 Last modified on 2013-03-22 17:40:58 Owner jirka (4157) Last modified by jirka (4157) Numerical id 5 Author jirka (4157) Entry type Definition Classification msc 58A10 Classification msc 53-00 Synonym volume form Synonym volume measure Defines area element Defines area form Defines area measure Defines Euclidean volume element Defines Euclidean volume form Defines euclidean volume measure Defines surface area measure Defines surface area element Defines surface area form