Inmathematics, avolume element provides a means forintegrating afunction with respect tovolume in various coordinate systems such asspherical coordinates andcylindrical coordinates. Thus a volume element is an expression of the form$${\displaystyle \mathrm{d}V=\rho (u_1,u_2,u_3)\mathrm{d}{u}_1\mathrm{d}{u}_2\mathrm{d}{u}_3}$$where the${\displaystyle u_i}$ are the coordinates, so that the volume of any set${\displaystyle B}$ can be computed by$${\displaystyle \mathrm{Volume}(B)={\int }_B\rho (u_1,u_2,u_3)\mathrm{d}{u}_1\mathrm{d}{u}_2\mathrm{d}{u}_3.}$$For example, in spherical coordinates${\displaystyle \mathrm{d}V=u_1^2\sin u_2\mathrm{d}{u}_1\mathrm{d}{u}_2\mathrm{d}{u}_3}$, and so${\displaystyle \rho =u_1^2\sin u_2}$.

The notion of a volume element is not limited to three dimensions: in two dimensions it is often known as thearea element, and in this setting it is useful for doingsurface integrals. Under changes of coordinates, the volume element changes by the absolute value of theJacobian determinant of the coordinate transformation (by thechange of variables formula). This fact allows volume elements to be defined as a kind ofmeasure on amanifold. On anorientabledifferentiable manifold, a volume element typically arises from avolume form: a top degreedifferential form. On a non-orientable manifold, the volume element is typically theabsolute value of a (locally defined) volume form: it defines a1-density.

InEuclidean space, the volume element is given by the product of the differentials of the Cartesian coordinates$${\displaystyle \mathrm{d}V=\mathrm{d}x\mathrm{d}y\mathrm{d}z.}$$In different coordinate systems of the form${\displaystyle x=x(u_1,u_2,u_3)}$,${\displaystyle y=y(u_1,u_2,u_3)}$,${\displaystyle z=z(u_1,u_2,u_3)}$, the volume elementchanges by the Jacobian (determinant) of the coordinate change:$${\displaystyle \mathrm{d}V=\left|\frac{\mathrm{\partial }(x,y,z)}{\mathrm{\partial }(u_1,u_2,u_3)}\right|\mathrm{d}{u}_1\mathrm{d}{u}_2\mathrm{d}{u}_3.}$$For example, in spherical coordinates (mathematical convention)the Jacobian determinant is$${\displaystyle \left|\frac{\mathrm{\partial }(x,y,z)}{\mathrm{\partial }(\rho ,\varphi ,\theta )}\right|={\rho }^2\sin \varphi }$$so that$${\displaystyle \mathrm{d}V={\rho }^2\sin \varphi \mathrm{d}\rho \mathrm{d}\theta \mathrm{d}\varphi .}$$This can be seen as a special case of the fact that differential forms transform through a pullback${\displaystyle F^{\ast }}$ as$${\displaystyle F^{\ast }(ud{y}^1\wedge \cdots \wedge d{y}^n)=(u\circ F)det\left(\frac{\mathrm{\partial }{F}^j}{\mathrm{\partial }{x}^i}\right)\mathrm{d}{x}^1\wedge \cdots \wedge \mathrm{d}{x}^n}$$

Consider thelinear subspace of then-dimensionalEuclidean spaceRⁿ that is spanned by a collection oflinearly independent vectors$${\displaystyle X_1,\dots ,X_k.}$$To find the volume element of the subspace, it is useful to know the fact fromlinear algebra that the volume of theparallelepiped spanned by the${\displaystyle X_i}$ is the square root of thedeterminant of theGramian matrix of the${\displaystyle X_i}$:$${\displaystyle \sqrt{det(X_i\cdot X_j{)}_{i,j=1\dots k}}.}$$

Any pointp in the subspace can be given coordinates${\displaystyle (u_1,u_2,\dots ,u_k)}$ such that$${\displaystyle p=u_1{X}_1+\cdots +u_k{X}_k.}$$At a pointp, if we form a small parallelepiped with sides${\displaystyle \mathrm{d}{u}_i}$, then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix$${\displaystyle \sqrt{det{\left((d{u}_i{X}_i)\cdot (d{u}_j{X}_j)\right)}_{i,j=1\dots k}}=\sqrt{det(X_i\cdot X_j{)}_{i,j=1\dots k}}\mathrm{d}{u}_1\mathrm{d}{u}_2\cdots \mathrm{d}{u}_k.}$$This therefore defines the volume form in the linear subspace.

On anorientedRiemannian manifold of dimensionn, the volume element is a volume form equal to theHodge dual of the unit constant function,${\displaystyle f(x)=1}$:$${\displaystyle \omega =\star 1.}$$Equivalently, the volume element is precisely theLevi-Civita tensor${\displaystyle ϵ}$.^[1] In coordinates,$${\displaystyle \omega =ϵ=\sqrt{\left|detg\right|}\mathrm{d}{x}^1\wedge \cdots \wedge \mathrm{d}{x}^n}$$where${\displaystyle detg}$ is thedeterminant of themetric tensorg written in the coordinate system.

A simple example of a volume element can be explored by considering a two-dimensional surface embedded inn-dimensionalEuclidean space. Such a volume element is sometimes called anarea element. Consider a subset${\displaystyle U\subset {\mathbb{R}}^2}$ and a mapping function$${\displaystyle \phi :U\to {\mathbb{R}}^n}$$thus defining a surface embedded in${\displaystyle {\mathbb{R}}^n}$. 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$${\displaystyle f(u_1,u_2)\mathrm{d}{u}_1\mathrm{d}{u}_2}$$that allows one to compute the area of a setB lying on the surface by computing the integral$${\displaystyle \mathrm{Area}(B)={\int }_{B}f(u_1,u_2)\mathrm{d}{u}_1\mathrm{d}{u}_2.}$$

Here we will find the volume element on the surface that defines area in the usual sense. TheJacobian matrix of the mapping is$${\displaystyle J_{ij}=\frac{\mathrm{\partial }{\phi }_i}{\mathrm{\partial }{u}_j}}$$with indexi running from 1 ton, andj running from 1 to 2. The Euclideanmetric in then-dimensional space induces a metric${\displaystyle g=J^{T}J}$ on the setU, with matrix elements$${\displaystyle g_{ij}=\sum _{k=1}^n{J}_{ki}{J}_{kj}=\sum _{k=1}^n\frac{\mathrm{\partial }{\phi }_k}{\mathrm{\partial }{u}_i}\frac{\mathrm{\partial }{\phi }_k}{\mathrm{\partial }{u}_j}.}$$

Thedeterminant of the metric is given by$${\displaystyle detg={\left|\frac{\mathrm{\partial }\phi }{\mathrm{\partial }{u}_1}\wedge \frac{\mathrm{\partial }\phi }{\mathrm{\partial }{u}_2}\right|}^2=det(J^{T}J)}$$

For a regular surface, this determinant is non-vanishing; equivalently, the Jacobian matrix has rank 2.

Now consider a change of coordinates onU, given by adiffeomorphism$${\displaystyle f:U\to U,}$$so that the coordinates${\displaystyle (u_1,u_2)}$ are given in terms of${\displaystyle (v_1,v_2)}$ by${\displaystyle (u_1,u_2)=f(v_1,v_2)}$. The Jacobian matrix of this transformation is given by$${\displaystyle F_{ij}=\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }{v}_j}.}$$

In the new coordinates, we have$${\displaystyle \frac{\mathrm{\partial }{\phi }_i}{\mathrm{\partial }{v}_j}=\sum _{k=1}^2\frac{\mathrm{\partial }{\phi }_i}{\mathrm{\partial }{u}_k}\frac{\mathrm{\partial }{f}_k}{\mathrm{\partial }{v}_j}}$$and so the metric transforms as$${\displaystyle \tilde{g}=F^{T}gF}$$where${\displaystyle \tilde{g}}$ is the pullback metric in thev coordinate system. The determinant is$${\displaystyle det\tilde{g}=detg{\left(detF\right)}^2.}$$

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${\displaystyle B\subset U}$ 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.

For example, consider the sphere with radiusr centered at the origin inR³. This can be parametrized usingspherical coordinates with the map$${\displaystyle \varphi (u_1,u_2)=(r\cos u_1\sin u_2,r\sin u_1\sin u_2,r\cos u_2).}$$Thenand the area element is$${\displaystyle \omega =\sqrt{detg}\mathrm{d}{u}_1\mathrm{d}{u}_2=r^2\sin u_2\mathrm{d}{u}_1\mathrm{d}{u}_2.}$$

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

• 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

• Measure theory
• Integral calculus
• Multivariable calculus

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