Definition:Interior Multiplication

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 Progress In 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 Divergence‎ (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 $\omega \mapsto i_v \omega$.

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Further research is required in order to fill out the details. In particular: Alternating forms You 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