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 1997).

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.

REFERENCES:

Gray, A. "Isometries and Conformal Maps of Surfaces." §15.2 in Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 346-351, 1997.