This is exercise 20.9 of Tu's "Introduction to Manifolds" ( 2nd edition).
Let $\omega = dx^1 \wedge \cdots \wedge dx^n $ be the volume form and $X=\sum x^i \partial/\partial{x^i}$ the radial vector field on $\mathbb{R}^n$. Compute the contraction ${\iota}_X\omega$.
For the computation I have applied the formula of the contraction of a wedge product of 1-covectors $\alpha^1 \wedge \cdots \wedge \alpha^n$
$\iota_X(\alpha^1 \wedge \cdots \wedge \alpha^n) = \sum_{i=1}^n (-1)^{i-1} \alpha^i(X) \alpha^1\wedge\alpha^2\wedge \cdots \wedge \widehat{\alpha^i} \wedge \cdots \wedge\alpha^n$ (where the caret element $\widehat{\alpha^i}$ means that $\alpha^i$ is excluded from the wedge)
so:
$$\iota_X\omega = \sum_{i=1}^n (-1)^{i-1}x^i \, dx^1\wedge dx^2 \cdots \wedge \widehat{dx^i}\wedge\cdots\wedge dx^n$$
My question is: can this $(n-1)$-form expression be developed further in a more coincise expression (that maybe was the aim of the exercise)? I can't see anything particular in this expression apart from the fact that the exterior derivative of each term $(-1)^{i-1} x^i \, dx^1\wedge dx^2 \wedge \cdots \wedge\widehat{dx^i} \wedge \cdots \wedge dx^n$ is again the initial volume form, so we have $d\iota_X\omega=n \omega$. Maybe I am missing something? thanks.