The exterior derivative of a $k$-form as a multilinear function

Question

Let $w$ be a $k$-form, then it can be viewed as a multilinear function of vector fields of the manifold. $dw$ is a $k+1$ form thus a multilinear function on vector fields as well. Given vector fields $X_1, \ldots, X_{k+1}$. I want to know what is $dw(X_1, \ldots, X_{k+1})$ in terms of $w$. This can be derived of course directly via the alternating tensor formulas and induction. However, the computation is sort of tedious. Is there an easier way to compute it?

Answer 1

I don't know if my answer is what you are looking for but:

$$d\varepsilon(X_1, \ldots, X_{k+1})=\sum_{\sigma\in \mathsf{Sh}(1, k)} \mathsf{sgn}(\sigma) X_{\sigma(1)} \varepsilon(X_{\sigma(2)}, \ldots, X_{\sigma(k+1)})+\sum_{\sigma\in\mathsf{Sh}(2, k-1)} \mathsf{sgn}(\sigma) \varepsilon([X_{\sigma(1)}, X_{\sigma(2)}], X_{\sigma(3)}, \ldots, X_{\sigma(k+1)})$$

where $\mathsf{Sh}(p, q)$ is the set of those permutations $\sigma$ of the set $\{1, 2, \ldots, p+q\}$ such that

$$\sigma(1)<\ldots<\sigma(p)\quad \textrm{and}\quad \sigma(p+1)<\ldots<\sigma(p+q).$$