Wedge product of differential forms ($\wedge$)

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:In particular, $dx\wedge dy=-dy\wedge dx$ and $dx\wedge dx=0$. For planar vectors $v,w$,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.

Exterior multiplication by a fixed differential form is the linear operator taking the wedge product of differential forms on the left. For a one-form $\omega$, it raises degree by one. Anticommutativity gives $e(\omega )e(\eta )+e(\eta )e(\omega )=0$ for one-forms. On the exterior algebra of an inner product space, pairing this operation with contraction produces Clifford multiplication $c(v)=e(v^{♭})-{\iota }_v$.