Volume element

Short description: Concept in integration theory

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 dV=ρ(u1,u2,u3)du1du2du3 where the ui are the coordinates, so that the volume of any set B can be computed by Volume⁡(B)=∫Bρ(u1,u2,u3)du1du2du3. For example, in spherical coordinates dV=u12sin⁡u2du1du2du3, and so ρ=u12sin⁡u2.

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.

## Contents

* 1 Volume element in Euclidean space
* 2 Volume element of a linear subspace
* 3 Volume element of manifolds
* 3.1 Area element of a surface
* 3.2 Example: Sphere
* 4 See also
* 5 References

## Volume element in Euclidean space

In Euclidean space, the volume element is given by the product of the differentials of the Cartesian coordinates dV=dxdydz. In different coordinate systems of the form x=x(u1,u2,u3), y=y(u1,u2,u3), z=z(u1,u2,u3), the volume element changes by the Jacobian (determinant) of the coordinate change: dV=|∂(x,y,z)∂(u1,u2,u3)|du1du2du3. For example, in spherical coordinates (mathematical convention) x=ρcos⁡θsin⁡ϕy=ρsin⁡θsin⁡ϕz=ρcos⁡ϕ the Jacobian determinant is |∂(x,y,z)∂(ρ,ϕ,θ)|=ρ2sin⁡ϕ so that dV=ρ2sin⁡ϕdρdθdϕ. This can be seen as a special case of the fact that differential forms transform through a pullback F* as F*(udy1∧⋯∧dyn)=(u∘F)det⁡(∂Fj∂xi)dx1∧⋯∧dxn

## 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 X1,…,Xk. 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 Xi is the square root of the determinant of the Gramian matrix of the Xi: det⁡(Xi⋅Xj)i,j=1…k.

Any point p in the subspace can be given coordinates (u1,u2,…,uk) such that p=u1X1+⋯+ukXk. At a point p, if we form a small parallelepiped with sides dui, then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix det⁡((duiXi)⋅(dujXj))i,j=1…k=det⁡(Xi⋅Xj)i,j=1…kdu1du2⋯duk. 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: ω=⋆1. Equivalently, the volume element is precisely the Levi-Civita tensor ϵ.[1] In coordinates, ω=ϵ=|det⁡g|dx1∧⋯∧dxn 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⊂ℝ2 and a mapping function φ:U->ℝn thus defining a surface embedded in ℝn. 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(u1,u2)du1du2 that allows one to compute the area of a set B lying on the surface by computing the integral Area⁡(B)=∫Bf(u1,u2)du1du2.

Here we will find the volume element on the surface that defines area in the usual sense. The Jacobian matrix of the mapping is Jij=∂φi∂uj 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=JTJ on the set U, with matrix elements gij=∑k=1nJkiJkj=∑k=1n∂φk∂ui∂φk∂uj.

The determinant of the metric is given by det⁡g=|∂φ∂u1∧∂φ∂u2|2=det⁡(JTJ)

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:U->U, so that the coordinates (u1,u2) are given in terms of (v1,v2) by (u1,u2)=f(v1,v2). The Jacobian matrix of this transformation is given by Fij=∂fi∂vj.

In the new coordinates, we have ∂φi∂vj=∑k=12∂φi∂uk∂fk∂vj and so the metric transforms as g~=FTgF where g~ is the pullback metric in the v coordinate system. The determinant is det⁡g~=det⁡g(det⁡F)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⊂U is given by the integral Area(B)=∬Bdet⁡gdu1du2=∬Bdet⁡g|det⁡F|dv1dv2=∬Bdet⁡g~dv1dv2.

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 ϕ(u1,u2)=(rcos⁡u1sin⁡u2,rsin⁡u1sin⁡u2,rcos⁡u2). Then g=(r2sin2u200r2), and the area element is ω=det⁡gdu1du2=r2sin⁡u2du1du2.

## See also

* Cylindrical coordinate system § Line and volume elements
* Spherical coordinate system § Integration and differentiation in spherical coordinates
* Volume integral
* Surface integral
* Line integral
* Line element

## References

* Besse, Arthur L. (1987), Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Berlin, New York: Springer-Verlag, pp. xii+510, ISBN 978-3-540-15279-8

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

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