Volume Element

A volume element is the differential element $dV$ whose Volume Integral over some range in a given coordinate system gives the Volume of a solid,

$$V=\int\!\!\!\int\!\!\!\int _G dx\,dy\,dz.$$ (1)

In $\Bbb{R}^n$, the volume of the infinitesimal $n$-Hypercube bounded by $dx_1$, ..., $dx_n$ has volume given by the Wedge Product

$$dV=dx_1\wedge \ldots\wedge dx_n$$ (2)

(Gray 1993).

The use of the antisymmetric Wedge Product instead of the symmetric product $dx_1\ldots dx_n$ is a technical refinement often omitted in informal usage. Dropping the wedges, the volume element for Curvilinear Coordinates in $\Bbb{R}^3$ is given by

$$dV = \vert(h_1{\hat {\bf u}}_1\,du_1)\cdot(h_2{\hat {\bf u}}_2\,du_2)\times (h_3{\hat {\bf u}}_3\,du_3)\vert$$ (3)

$$= h_1h_2h_3\,du_1\,du_2\,du_3$$ (4)

$$= \left\vert{{\partial{\bf r}\over\partial u_1}\cdot {\partial{\bf ... ...ial u_2} \times{\partial{\bf r}\over\partial u_3}}\right\vert\,du_1\,du_2\,du_3$$ (5)

$$= \left\vert\begin{array}{ccc}{\partial x\over\partial u_1} & {\par... ...al u_2} & {\partial z\over\partial u_3}\end{array}\right\vert\,du_1\,du_2\,du_3$$ (6)

$$= \left\vert{{\partial (x,y,z)\over \partial (u_1,u_2,u_3)}}\right\vert\,du_1\,du_2\,du_3,$$ (7)

where the latter is the Jacobian and the $h_i$ are Scale Factors.

See also Area Element, Jacobian, Line Element, Riemannian Metric, Scale Factor, Surface Integral, Volume Integral

References

Gray, A. ``Isometries of Surfaces.'' §13.2 in Modern Differential Geometry of Curves and Surfaces. Boca Raton, FL: CRC Press, pp. 255-258, 1993.