Prove $\text{div(rot}X)=0$, $\forall X\in \mathcal{X(\mathbb{R^n})}$
$X$ is a vector field on $\mathbb{R^n}$, $\omega$ it's a form
$X\mapsto \omega \mapsto d\omega \mapsto *(d\omega)=\text{rot}X$
(problem 14 from do Carmo's differential forms.)
$\text{rot}X$ is an $(n-2)$-form which maintains coefficients, so $\text{rot}X= \sum_I a_Idx^I$, here is where I stuck I don't know how to compute the $\text{div}$ of an $(n-2)$-form.
For example, in $\mathbb{R^4}$ $\text{rot}X=a_{12}dx^1\wedge dx^2+a_{13}dx^1\wedge dx^3+a_{14}dx^1\wedge dx^4+a_{23}dx^2\wedge dx^3+a_{24}dx^2\wedge dx^4+a_{34}dx^3\wedge dx^4$ how do I compute $\text{div(rot}X)$ ?