= Wedge product of differential forms
{title2=$\wedge$}
= Wedge products of differential forms
{synonym}
The wedge product combines a degree-$p$ <differential form> and a degree-$q$ <differential form> into one of degree $p+q$. It is <associative> and graded-commutative:
$$
\alpha\wedge\beta=(-1)^{pq}\beta\wedge\alpha.
$$
In particular, $dx\wedge dy=-dy\wedge dx$ and $dx\wedge dx=0$. For planar <vectors> $v,w$,
$$
(dx\wedge dy)(v,w)=v_xw_y-v_yw_x.
$$
This <determinant> is the signed area of their <parallelogram>. Exchanging the edges reverses the sign; parallel edges give zero. Thus antisymmetry records the geometry of oriented area rather than adding an arbitrary algebraic convention. The <wedge product of differential forms> requires no <Riemannian metric> to define it. It obeys the <graded Leibniz rule> under the <exterior derivative>.
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