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