Volume Element
المؤلف:
Gray, A
المصدر:
"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
الجزء والصفحة:
pp. 346-351
17-9-2018
2565
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
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.
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