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
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
For example, in spherical coordinates (mathematical convention)
= ρ cos θ sin ϕ
This can be seen as a special case of the fact that differential forms transform through a pullback
Consider the linear subspace of the n-dimensional Euclidean space Rn that is spanned by a collection of linearly independent vectors
To find the volume element of the subspace, it is useful to know the fact from linear algebra that the volume of the parallelepiped spanned by the
is the square root of the determinant of the Gramian matrix of the
At a point p, if we form a small parallelepiped with sides
, then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix
This therefore defines the volume form in the linear subspace.
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,
Equivalently, the volume element is precisely the Levi-Civita tensor
is the determinant of the metric tensor g written in the coordinate system.
A simple example of a volume element can be explored by considering a two-dimensional surface embedded in n-dimensional Euclidean space.
In two dimensions, volume is just area, and a volume element gives a way to determine the area of parts of the surface.
Thus a volume element is an expression of the form
Here we will find the volume element on the surface that defines area in the usual sense.
For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2.
is the pullback metric in the v coordinate system.
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.
For example, consider the sphere with radius r centered at the origin in R3.
This can be parametrized using spherical coordinates with the map