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In mathematics, a **volume element** provides a means for integrating a function with respect to volume in various coordinate systems such as spherical coordinates and cylindrical coordinates. Thus a volume element is an expression of the form

where the are the coordinates, so that the volume of any set can be computed by

For example, in spherical coordinates , and so .

The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as the **area element**, and in this setting it is useful for doing surface integrals. Under changes of coordinates, the volume element changes by the absolute value of the Jacobian determinant of the coordinate transformation (by the change of variables formula). This fact allows volume elements to be defined as a kind of measure on a manifold. On an orientable differentiable manifold, a volume element typically arises from a volume form: a top degree differential form. On a non-orientable manifold, the volume element is typically the absolute value of a (locally defined) volume form: it defines a 1-density.

In Euclidean space, the volume element is given by the product of the differentials of the Cartesian coordinates

Consider the linear subspace of the *n*-dimensional Euclidean space **R**^{n} that is spanned by a collection of linearly independent vectors

Any point *p* in the subspace can be given coordinates such that

On an *oriented* Riemannian manifold of dimension *n*, the volume element is a volume form equal to the Hodge dual of the unit constant function, :

It has been suggested that this section be split out into another article titled Surface integral. (Discuss) (December 2023) |

A simple example of a volume element can be explored by considering a two-dimensional surface embedded in *n*-dimensional Euclidean space. Such a volume element is sometimes called an *area element*. Consider a subset and a mapping function

Here we will find the volume element on the surface that defines area in the usual sense. The Jacobian matrix of the mapping is

The determinant of the metric is given by

For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2.

Now consider a change of coordinates on *U*, given by a diffeomorphism

In the new coordinates, we have

Given the above construction, it should now be straightforward to understand how the volume element is invariant under an orientation-preserving change of coordinates.

In two dimensions, the volume is just the area. The area of a subset is given by the integral

Thus, in either coordinate system, the volume element takes the same expression: the expression of the volume element is invariant under a change of coordinates.

Note that there was nothing particular to two dimensions in the above presentation; the above trivially generalizes to arbitrary dimensions.

For example, consider the sphere with radius *r* centered at the origin in **R**^{3}. This can be parametrized using spherical coordinates with the map

- Besse, Arthur L. (1987),
*Einstein manifolds*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: Springer-Verlag, pp. xii+510, ISBN 978-3-540-15279-8

**^**Carroll, Sean.*Spacetime and Geometry*. Addison Wesley, 2004, p. 90