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.
