volume element

If M is an n dimensional manifoldMathworldPlanetmath, 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, ω or voln. If the manifold is a Riemannian manifoldMathworldPlanetmath with g, then the natural volume form is defined in local coordinates x1xn 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 n, then the usual volume element dV=dx1dx2dxn is called the Euclidean volume element or Euclidean volume form. In this context, n is usually treated as 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 submanifoldMathworldPlanetmath of 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…


  • 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