# Definition:Interior Multiplication

Jump to navigation
Jump to search

## 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 $v \mathop \lrcorner \omega$ 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:

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

Work In ProgressIn particular: Review the notation $\lrcorner$ and decide whether $v \rfloor \omega$ is "better" -- the latter has already been used on the non-compliant page Coordinate Representation of Divergenceā (linkless)You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by completing it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{WIP}}` from the code. |

## Also denoted as

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

This article needs to be linked to other articles.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{MissingLinks}}` from the code. |

Further research is required in order to fill out the details.In particular: Alternating formsYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Research}}` from the code. |

## Sources

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