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,
| [画像: 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)
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 IntegralExplore with Wolfram|Alpha
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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 ElementCite this as:
Weisstein, Eric W. "Volume Element." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/VolumeElement.html