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,


In 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 ^ ... ^ dx_n

(Gray 1997).

The use of the antisymmetric wedge product instead of the symmetric product dx_1...dx_n is a technical refinement often omitted in informal usage. Dropping the wedges, the volume element for curvilinear coordinates in R^3 is given by

=|(partialx)/(partialu_1) (partialx)/(partialu_2) (partialx)/(partialu_3); (partialy)/(partialu_1) (partialy)/(partialu_2) (partialy)/(partialu_3); (partialz)/(partialu_1) (partialz)/(partialu_2) (partialz)/(partialu_3)|du_1du_2du_3

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 Area, Surface Integral, Volume Integral

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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.

Referenced on Wolfram|Alpha

Volume Element

Cite this as:

Weisstein, Eric W. "Volume Element." From MathWorld--A Wolfram Web Resource.

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