# Definition:Interior Multiplication

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## Definition

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

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

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

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

- $v\, \lrcorner\, \omega := 0$

## Also denoted as

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

## Sources

- 2018: John M. Lee:
*Introduction to Riemannian Manifolds*(2nd ed.): Appendix $\text B$. Review of Tensors