(Spivak 1999, p. 203), where and are constants. The exterior
algebra is generated by elements of degree one, and so the wedge product can
be defined using a basis
for :

(5)

when the indices
are distinct, and the product is zero otherwise.

While the formula
holds when
has degree one, it does not hold in general. For example, consider :

(6)

(7)

(8)

If have degree one, then
they are linearly independent iff .

The wedge product is the "correct" type of product to use in computing
a volume element

(9)

The wedge product can therefore be used to calculate determinants and volumes of parallelepipeds. For example, write
where are the columns of . Then