The interior product (also known as interior derivative, interior multiplication, inner multiplication, inner derivative, insertion operator, or inner derivation) is a degree −1 (anti)derivation on the exterior algebra of differential forms on a smooth manifold. The interior product, named in opposition to the exterior product, should not be confused with an inner product .
It is usually denoted by $i_X \omega$ or $X\,\lrcorner\,\omega$ for $X\in \mathfrak{X}(M)$ and $\omega \in \Omega^1 (M)$.
It is an antiderivation of degree -1 so
$$ \iota_{X}(\beta \wedge \gamma)=\left(\iota_{X} \beta\right) \wedge \gamma+(-1)^{p} \beta \wedge\left(\iota_{X} \gamma\right). $$where $\beta$ is a $p$-form and $\gamma$ a $q$-form.
Other expressions:
See also formulas for Lie derivative, exterior derivatives, bracket, interior product.
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Author of the notes: Antonio J. Pan-Collantes
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