Let \(V\) be a finite dimensional vector space. Then the interior multiplication \(i_v\colon \Lambda^k(V^*)\to \Lambda^{k-1}(V^*)\) relates a vector \(v\) with an alternating tensor . It is defined by
\begin{equation*} (i_v \omega)(w_1,\ldots,w_{k-1})= \omega(v,w_1,\ldots,w_{k-1}). \end{equation*}Notation: \(v \lrcorner \omega\)
Remarks
- \(i_v\circ i_v = 0\), since alternating tensors vanish if two arguments are equal.
- Applying the interior multiplication on the wedge product of two alternating tensors \(\omega\in \Lambda^k(V^*)\) and \(\eta\in \Lambda^l(V^*)\) yields \begin{equation*} i_v(\omega\wedge \eta)=(i_v\omega)\wedge \eta + (-1)^k \omega\wedge (i_v\eta). \end{equation*}