Exterior multiplication ($e(\omega )\alpha =\omega \wedge \alpha$)

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 $c(v)=e(v^{♭})-{\iota }_v$.