Volume element

In Euclidean space, the volume element is given by the product of the differentials of the Cartesian coordinates \mathrm{d}V = \mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z. 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: \mathrm{d}V = \left|\frac{\partial (x,y,z)}{\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) \begin{align} x &= \rho \cos\theta \sin\phi\\ y &= \rho \sin\theta \sin\phi\\ z &= \rho \cos\phi \end{align} the Jacobian determinant is \left |\frac{\partial(x,y,z)}{\partial (\rho,\phi,\theta)}\right| = \rho^2\sin\phi so that \mathrm{d}V = \rho^2\sin\phi\,\mathrm{d}\rho\,\mathrm{d}\theta\,\mathrm{d}\phi. This can be seen as a special case of the fact that differential forms transform through a pullback F^* as F^*(u \; dy^1 \wedge \cdots \wedge dy^n) = (u \circ F) \det \left(\frac{\partial F^j}{\partial x^i}\right) \mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^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,\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: \sqrt{\det(X_i\cdot X_j)_{i,j=1\dots k}}. Any point p in the subspace can be given coordinates (u_1,u_2,\dots,u_k) such that p = u_1X_1 + \cdots + u_kX_k. At a point p, if we form a small parallelepiped with sides \mathrm{d}u_i, then the volume of that parallelepiped is the square root of the determinant of the Grammian matrix \sqrt{\det\left((du_i X_i)\cdot (du_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. ==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: \omega = \star 1 . Equivalently, the volume element is precisely the Levi-Civita tensor \epsilon. In coordinates, \omega = \epsilon =\sqrt{\left|\det g\right|}\, \mathrm{d}x^1 \wedge \cdots \wedge \mathrm{d}x^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 \varphi:U\to \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)\,\mathrm{d}u_1\,\mathrm{d}u_2 that allows one to compute the area of a set B lying on the surface by computing the integral \operatorname{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. The Jacobian matrix of the mapping is J_{ij} = \frac{\partial \varphi_i} {\partial u_j} 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_{ij}=\sum_{k=1}^n J_{ki} J_{kj} = \sum_{k=1}^n \frac{\partial \varphi_k} {\partial u_i} \frac{\partial \varphi_k} {\partial u_j}. The determinant of the metric is given by \det g = \left| \frac{\partial \varphi} {\partial u_1} \wedge \frac{\partial \varphi} {\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 on U, given by a diffeomorphism f \colon U\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_{ij}= \frac{\partial f_i} {\partial v_j}. In the new coordinates, we have \frac{\partial \varphi_i} {\partial v_j} = \sum_{k=1}^2 \frac{\partial \varphi_i} {\partial u_k} \frac{\partial f_k} {\partial v_j} and so the metric transforms as \tilde{g} = F^T g F where \tilde{g} is the pullback metric in the v coordinate system. The determinant is \det \tilde{g} = \det g \left( \det F \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 \begin{align} \mbox{Area}(B) &= \iint_B \sqrt{\det g}\; \mathrm{d}u_1\; \mathrm{d}u_2 \\[1.6ex] &= \iint_B \sqrt{\det g} \left|\det F\right| \;\mathrm{d}v_1 \;\mathrm{d}v_2 \\[1.6ex] &= \iint_B \sqrt{\det \tilde{g}} \;\mathrm{d}v_1 \;\mathrm{d}v_2. \end{align} 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 \phi(u_1,u_2) = (r \cos u_1 \sin u_2, r \sin u_1 \sin u_2, r \cos u_2). Then g = \begin{pmatrix} r^2\sin^2u_2 & 0 \\ 0 & r^2 \end{pmatrix}, and the area element is \omega = \sqrt{\det g}\; \mathrm{d}u_1 \mathrm{d}u_2 = r^2\sin u_2\, \mathrm{d}u_1 \mathrm{d}u_2. ==See also==