Volume element on arvau

Wikipedia / Volume element

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.

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.

Volume element in Euclidean space

In Euclidean space, the volume element is given by the product of the differentials of the Cartesian coordinates, the volume element changes by the absolute value of the Jacobian determinant For example, in spherical coordinates

Volume element of a linear subspace

Consider the linear subspace of the n-dimensional Euclidean space Rn that is spanned by a collection of linearly independent To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped is the square root of the determinant of the Gramian matrix

Any point p At a point p This therefore defines the volume form in the linear subspace.

Volume element of manifolds

On an oriented Riemannian manifold of dimension n, the volume element is a volume form equal to the Hodge dual Equivalently, the volume element is precisely the Levi-Civita tensor. is the determinant of the metric tensor g written in the coordinate system.

Area element of a surface

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 that allows one to compute the area of a set B

Here we will find the volume element on the surface that defines area in the usual sense. The Jacobian matrix with index i running from 1 to n, and j running from 1 to 2. The Euclidean metric in the n on the set U

The determinant

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

is the pullback metric in the v

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

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.

Example: Sphere

For example, consider the sphere with radius r centered at the origin in R3. This can be parametrized using spherical coordinates