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. Consider a subset and a mapping function
thus defining a surface embedded in . 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
that allows one to compute the area of a set B lying on the surface by computing the integral
Here we will find the volume element on the surface that defines area in the usual sense. The Jacobian matrix of the mapping is
with index i running from 1 to n, and j running from 1 to 2. The Euclidean metric in the n-dimensional space induces a metric on the set U, with matrix elements
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
so that the coordinates are given in terms of by . The Jacobian matrix of this transformation is given by
In the new coordinates, we have
and so the metric transforms as
where is the pullback metric in the v coordinate system. The determinant is
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.