## Volume Element

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

 (1)

In , the volume of the infinitesimal -Hypercube bounded by , ..., has volume given by the Wedge Product
 (2)

(Gray 1993).

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

 (3) (4) (5) (6) (7)

where the latter is the Jacobian and the 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.