Interior product or contraction

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 $X⌟\omega$ for $X\in \mathfrak{X}(M)$ and $\omega \in {\mathrm{\Omega }}^1(M)$.

It is an antiderivation of degree -1 so

where $\beta$ is a $p$-form and $\gamma$ a $q$-form.
Other expressions:

• ${\iota }_{X}f=0$
• ${\iota }_{X}df=X(f)$
• ${\iota }_{X_1\wedge \cdots \wedge X_n}\omega =\omega (X_1,\dots ,X_n)$.
• Cartan formula
• $d{i}_X\omega (Y)=[{\mathcal{L}}_X,i_Y]\omega +\omega ([X,Y])+Y\omega (X)$ , by Cartan formula.

See also formulas for Lie derivative, exterior derivatives, bracket, interior product.