the unit constant function, : Equivalently, the volume element is precisely the Levi-Civita tensor . In coordinates, where 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. 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. Example: Sphere For example, consider the sphere with radius r centered at the origin in R3. This can be parametrized using spherical coordinates with the map Then and the area element is See also Surface integral Volume integral References Measure theory Integral calculus Multivariable calculus