Volume element explained

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$$\backslash mathrmV\; =\; \backslash rho(u\_1,u\_2,u\_3)\backslash ,\backslash mathrmu\_1\backslash ,\backslash mathrmu\_2\backslash ,\backslash mathrmu\_3$$where the

u _{i} are the coordinates, so that the volume of any set B can be computed by$$\backslash operatorname(B)\; =\; \backslash int\_B\; \backslash rho(u\_1,u\_2,u\_3)\backslash ,\backslash mathrmu\_1\backslash ,\backslash mathrmu\_2\backslash ,\backslash mathrmu\_3.$$For example, in spherical coordinates d V = u ^{2 \sin} _{1} u _{2du} _{1du} _{2du} _{3} , and so \rho = u ^{2 \sin} _{1} u _{2} .

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$$\backslash mathrmV\; =\; \backslash mathrmx\backslash ,\backslash mathrmy\backslash ,\backslash mathrmz.$$In different coordinate systems of the form

x = x(u _{1,u} _{2,u} _{3)} , y = y(u _{1,u} _{2,u} _{3)} , z = z(u _{1,u} _{2,u} _{3)} , the volume element changes by the Jacobian (determinant) of the coordinate change:$$\backslash mathrmV\; =\; \backslash left|\backslash frac\backslash right|\backslash ,\backslash mathrmu\_1\backslash ,\backslash mathrmu\_2\backslash ,\backslash mathrmu\_3.$$For example, in spherical coordinates (mathematical convention) the Jacobian determinant is$$\backslash left\; |\backslash frac\backslash right|\; =\; \backslash rho^2\backslash sin\backslash phi$$so that$$\backslash mathrmV\; =\; \backslash rho^2\backslash sin\backslash phi\backslash ,\backslash mathrm\backslash rho\backslash ,\backslash mathrm\backslash theta\backslash ,\backslash mathrm\backslash phi.$$This can be seen as a special case of the fact that differential forms transform through a pullback F ^{*} as$$F^*(u\; \backslash ;\; dy^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dy^n)\; =\; (u\; \backslash circ\; F)\; \backslash det\; \backslash left(\backslash frac\backslash right)\; \backslash mathrmx^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; \backslash mathrmx^n$$

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 vectors$$X\_1,\backslash dots,X\_k.$$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

X _{i} is the square root of the determinant of the Gramian matrix of the X _{i} :$$\backslash sqrt.$$

Any point p in the subspace can be given coordinates

(u _{1,u} _{2,...,u} _{k)} such that$$p\; =\; u\_1X\_1\; +\; \backslash cdots\; +\; u\_kX\_k.$$At a point p, if we form a small parallelepiped with sides d u _{i} , then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix$$\backslash sqrt\; =\; \backslash sqrt\backslash ;\; \backslash mathrmu\_1\backslash ,\backslash mathrmu\_2\backslash ,\backslash cdots\backslash ,\backslash mathrmu\_k.$$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 of the unit constant function,

f(x) = 1 :$$\backslash omega\; =\; \backslash star\; 1\; .$$Equivalently, the volume element is precisely the Levi-Civita tensor \epsilon .^[1] In coordinates,$$\backslash omega\; =\; \backslash epsilon\; =\backslash sqrt\backslash ,\; \backslash mathrmx^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; \backslash mathrmx^n$$where \det g 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

U \subset \R ^{2} and a mapping function$$\backslash varphi:U\backslash to\; \backslash R^n$$thus defining a surface embedded in \R ^{n} . 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$$f(u\_1,u\_2)\backslash ,\backslash mathrmu\_1\backslash ,\backslash mathrmu\_2$$that allows one to compute the area of a set B lying on the surface by computing the integral$$\backslash operatorname(B)\; =\; \backslash int\_B\; f(u\_1,u\_2)\backslash ,\backslash mathrmu\_1\backslash ,\backslash mathrmu\_2.$$

Here we will find the volume element on the surface that defines area in the usual sense. The Jacobian matrix of the mapping is$$J\_\; =\; \backslash frac$$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

g = J ^{T} J on the set U, with matrix elements$$g\_=\backslash sum\_^n\; J\_\; J\_=\; \backslash sum\_^n\backslash frac\; \backslash frac\; .$$

The determinant of the metric is given by$$\backslash det\; g\; =\; \backslash left|\backslash frac\; \backslash wedge\backslash frac\; \backslash right|^2\; =\; \backslash det\; (J^T\; J)$$

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$$f\; \backslash colon\; U\backslash to\; U,$$so that the coordinates

(u _{1,} u _{2)} are given in terms of (v _{1,v} _{2)} by (u _{1,u} _{2)} = f(v _{1,v} _{2)} . The Jacobian matrix of this transformation is given by$$F\_=\; \backslash frac\; .$$

In the new coordinates, we have$$\backslash frac\; =\backslash sum\_^2\backslash frac\; \backslash frac$$and so the metric transforms as$$\backslash tilde\; =\; F^T\; g\; F$$where

\tilde{g} is the pullback metric in the v coordinate system. The determinant is$$\backslash det\; \backslash tilde\; =\; \backslash det\; g\; \backslash left(\backslash det\; F\; \backslash right)^2.$$

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

B\subset U 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$$\backslash phi(u\_1,u\_2)\; =\; (r\; \backslash cos\; u\_1\; \backslash sin\; u\_2,\; r\; \backslash sin\; u\_1\; \backslash sin\; u\_2,\; r\; \backslash cos\; u\_2).$$Thenand the area element is$$\backslash omega\; =\; \backslash sqrt\backslash ;\; \backslash mathrmu\_1\; \backslash mathrmu\_2\; =\; r^2\backslash sin\; u\_2\backslash ,\; \backslash mathrmu\_1\; \backslash mathrmu\_2.$$

See also

• • • Volume integral
• Surface integral
• Line integral
• Line element

Notes and References

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