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
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
where the latter is the Jacobian and the 
are scale factors.
