Interior product

The interior product is a way to go “down” a degree of a differential form (As opposed to the exterior derivative which goes “up” a degree). Thus, for a vector field $X$, ${\iota }_X:{\Omega }^p⟶{\Omega }^{p-1}.$

It is defined by holding the first spot of the differential form constant by the vector field and leaving all the others to vary. More formally, ${\iota }_X\alpha =\alpha (X,-,\dots ,-)$ or even more formally, ${\iota }_X\alpha (X_1,\dots ,X_{p-1})=\alpha (X,X_1,\dots ,X_{p-1}).$

Notation

Some authors use the notation $X┘\alpha ={\iota }_X\alpha$