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)
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In , the volume of the infinitesimal -hypercube bounded by , ..., has volume given by the wedge product
(2)
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(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)
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(4)
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(5)
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(6)
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(7)
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where the latter is the Jacobian and the are scale factors.