Definition:Interior Multiplication

From ProofWiki
Jump to navigation Jump to search


Let $V$ and $V^*$ be a vector space and its dual.

Let $\omega \in \map {\Lambda^k} {V^*}$ be an alternating $k$-tensor.

Let $v \in V$ be a vector.

Let $v \mathop \lrcorner \omega$ be an alternating $\paren {k - 1}$-tensor such that:

$\map {\paren {v \mathop \lrcorner \omega}} {w_1, \ldots, w_{k - 1}} := \map \omega {v, w_1, \ldots, w_{k - 1}}$

Then the mapping $\omega \mapsto v \mathop \lrcorner \omega$ is known as the interior multiplication by $v$.

If $\omega$ is a $0$-tensor then:

$v \mathop \lrcorner \omega := 0$

Also denoted as

The interior multiplication by $v$ is also denoted by $\omega \mapsto i_v \omega$.