A volume element is the differential element whose Volume Integral over some range in a given coordinate system gives the
Volume of a solid,
The use of the antisymmetric Wedge Product instead of the symmetric product
is a technical
refinement often omitted in informal usage. Dropping the wedges, the volume element for Curvilinear Coordinates in
is given by
See also Area Element, Jacobian, Line Element, Riemannian Metric, Scale Factor, Surface Integral, Volume Integral
Gray, A. ``Isometries of Surfaces.'' §13.2 in Modern Differential Geometry of Curves and Surfaces.
Boca Raton, FL: CRC Press, pp. 255-258, 1993.