Volume Element
A volume element is the differential element 
whose Volume Integral over some range in a given coordinate system gives theVolume of a solid,
 | (1) |
, the volume of the infinitesimal 
-Hypercube bounded by 
, ...,
has volume given by the Wedge Product | (2) |
The use of the antisymmetric Wedge Product instead of the symmetric product 
is a technicalrefinement 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.
- Reidemeister Moves
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