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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,

 V=intintint_(G)dxdydz.
(1)

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
(2)

(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

dV=|(h_1u_1^^du_1)·(h_2u_2^^du_2)x(h_3u_3^^du_3)|
(3)
=h_1h_2h_3du_1du_2du_3
(4)
=|(partialr)/(partialu_1)·(partialr)/(partialu_2)x(partialr)/(partialu_3)|du_1du_2du_3
(5)
=|(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
(6)
=|(partial(x,y,z))/(partial(u_1,u_2,u_3))|du_1du_2du_3,
(7)

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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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.

Referenced on Wolfram|Alpha

Volume Element

Cite this as:

Weisstein, Eric W. "Volume Element." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/VolumeElement.html

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