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
:$dV\; =\; \backslash rho(u\_1,u\_2,u\_3)\backslash ,du\_1\backslash ,du\_2\backslash ,du\_3$
where the $u\_i$ are the coordinates, so that the volume of any set $B$ can be computed by
:$\backslash operatorname(B)\; =\; \backslash int\_B\; \backslash rho(u\_1,u\_2,u\_3)\backslash ,du\_1\backslash ,du\_2\backslash ,du\_3.$
For example, in spherical coordinates $dV\; =\; u\_1^2\backslash sin\; u\_2\backslash ,du\_1\backslash ,du\_2\backslash ,du\_3$, and so $\backslash rho\; =\; u\_1^2\backslash sin\; u\_2$.
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 :$dV\; =\; dx\backslash ,dy\backslash ,dz.$ 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: :$dV\; =\; \backslash left|\backslash frac\backslash \backslash ,du\_1\backslash ,du\_2\backslash ,du\_3.$ For example, in spherical coordinates (mathematical convention) :$\backslash begin\; x\&=\backslash rho\backslash cos\backslash theta\backslash sin\backslash phi\backslash \backslash \; y\&=\backslash rho\backslash sin\backslash theta\backslash sin\backslash phi\backslash \backslash \; z\&=\backslash rho\backslash cos\backslash phi\; \backslash end$ the Jacobian determinant is :$\backslash left\; |\backslash frac\backslash \; =\; \backslash rho^2\backslash sin\backslash phi$ so that :$dV\; =\; \backslash rho^2\backslash sin\backslash phi\backslash ,d\backslash rho\backslash ,d\backslash theta\backslash ,d\backslash phi.$ This can be seen as a special case of the fact that differential forms transform through a pullback $F^*$ as :$F^*(u\; \backslash ;\; dy^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dy^n)\; =\; (u\; \backslash circ\; F)\; \backslash det\; \backslash left(\backslash frac\backslash right)\; dx^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^n$

** Volume element of a linear subspace **

Consider the linear subspace of the ''n''-dimensional Euclidean space R^{''n''} 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 $(u\_1,u\_2,\backslash dots,u\_k)$ such that
:$p\; =\; u\_1X\_1\; +\; \backslash cdots\; +\; u\_kX\_k.$
At a point ''p'', if we form a small parallelepiped with sides $du\_i$, then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix
:$\backslash sqrt\; =\; \backslash sqrt\backslash ;\; du\_1\backslash ,du\_2\backslash ,\backslash cdots\backslash ,du\_k.$
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 $\backslash epsilon$.Carroll, Sean. ''Spacetime and Geometry''. Addison Wesley, 2004, p. 90 In coordinates, :$\backslash omega\; =\; \backslash epsilon\; =\backslash sqrt\backslash ,\; dx^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^n$ where $\backslash det\; g$ 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 $U\; \backslash subset\; \backslash R^2$ and a mapping function
:$\backslash varphi:U\backslash to\; \backslash R^n$
thus defining a surface embedded in $\backslash R^n$. 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
:$f(u\_1,u\_2)\backslash ,du\_1\backslash ,du\_2$
that allows one to compute the area of a set ''B'' lying on the surface by computing the integral
:$\backslash operatorname(B)\; =\; \backslash int\_B\; f(u\_1,u\_2)\backslash ,du\_1\backslash ,du\_2.$
Here we will find the volume element on the surface that defines area in the usual sense. The Jacobian matrix of the mapping is
:$\backslash lambda\_=\backslash frac$
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 $g\; =\; \backslash lambda^T\; \backslash lambda$ on the set ''U'', with matrix elements
:$g\_=\backslash sum\_^n\; \backslash lambda\_\; \backslash lambda\_\; =\; \backslash sum\_^n\; \backslash frac\; \backslash frac\; .$
The determinant of the metric is given by
:$\backslash det\; g\; =\; \backslash left|\; \backslash frac\; \backslash wedge\; \backslash frac\; \backslash ^2\; =\; \backslash det\; (\backslash lambda^T\; \backslash lambda)$
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 $\backslash tilde$ 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 $B\backslash subset\; U$ is given by the integral
:$\backslash begin\; \backslash mbox(B)\; \&=\; \backslash iint\_B\; \backslash sqrt\backslash ;\; du\_1\backslash ;\; du\_2\; \backslash \backslash \; \&=\; \backslash iint\_B\; \backslash sqrt\; \backslash left|\backslash det\; F\backslash \; \backslash ;dv\_1\; \backslash ;dv\_2\; \backslash \backslash \; \&=\; \backslash iint\_B\; \backslash sqrt\; \backslash ;dv\_1\; \backslash ;dv\_2.\; \backslash end$
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 R^{3}. This can be parametrized using spherical coordinates with the map
:$\backslash phi(u\_1,u\_2)\; =\; (r\; \backslash cos\; u\_1\; \backslash sin\; u\_2,\; r\; \backslash sin\; u\_1\; \backslash sin\; u\_2,\; r\; \backslash cos\; u\_2).$
Then
:$g\; =\; \backslash begin\; r^2\backslash sin^2u\_2\; \&\; 0\; \backslash \backslash \; 0\; \&\; r^2\; \backslash end,$
and the area element is
:$\backslash omega\; =\; \backslash sqrt\backslash ;\; du\_1\; du\_2\; =\; r^2\backslash sin\; u\_2\backslash ,\; du\_1\; du\_2.$

See also

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

References

* {{reflist Category:Measure theory Category:Integral calculus Category:Multivariable calculus

Volume element in Euclidean space

In Euclidean space, the volume element is given by the product of the differentials of the Cartesian coordinates :$dV\; =\; dx\backslash ,dy\backslash ,dz.$ 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: :$dV\; =\; \backslash left|\backslash frac\backslash \backslash ,du\_1\backslash ,du\_2\backslash ,du\_3.$ For example, in spherical coordinates (mathematical convention) :$\backslash begin\; x\&=\backslash rho\backslash cos\backslash theta\backslash sin\backslash phi\backslash \backslash \; y\&=\backslash rho\backslash sin\backslash theta\backslash sin\backslash phi\backslash \backslash \; z\&=\backslash rho\backslash cos\backslash phi\; \backslash end$ the Jacobian determinant is :$\backslash left\; |\backslash frac\backslash \; =\; \backslash rho^2\backslash sin\backslash phi$ so that :$dV\; =\; \backslash rho^2\backslash sin\backslash phi\backslash ,d\backslash rho\backslash ,d\backslash theta\backslash ,d\backslash phi.$ This can be seen as a special case of the fact that differential forms transform through a pullback $F^*$ as :$F^*(u\; \backslash ;\; dy^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dy^n)\; =\; (u\; \backslash circ\; F)\; \backslash det\; \backslash left(\backslash frac\backslash right)\; dx^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^n$

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 $\backslash epsilon$.Carroll, Sean. ''Spacetime and Geometry''. Addison Wesley, 2004, p. 90 In coordinates, :$\backslash omega\; =\; \backslash epsilon\; =\backslash sqrt\backslash ,\; dx^1\; \backslash wedge\; \backslash cdots\; \backslash wedge\; dx^n$ where $\backslash det\; g$ is the determinant of the metric tensor ''g'' written in the coordinate system.

Example: Sphere

For example, consider the sphere with radius ''r'' centered at the origin in R

See also

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

References

* {{reflist Category:Measure theory Category:Integral calculus Category:Multivariable calculus