# 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 ${\mathrm{vol}}_{n}.$
If the manifold is a Riemannian manifold^{} with $g,$ then the natural volume form is defined in local coordinates ${x}^{1}\mathrm{\dots}{x}^{n}$ by

$$dV:=\sqrt{|g|}d{x}^{1}\wedge \mathrm{\dots}\wedge d{x}^{n}.$$ |

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 $dV=d{x}^{1}\wedge d{x}^{2}\wedge \mathrm{\dots}\wedge d{x}^{n}$ is called the Euclidean volume element or Euclidean volume form. In this context, ${\u2102}^{n}$ 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 |