I'm trying to prove the exact same formula as in this question:
$U \subset \mathbb{R^n}$ an open set, $\omega$ a k-differential form on $U$, $X_0, \dots, X_k$ vector fields on U, i. e. elements of $C^{\infty}(U)^n$.
I am supposed to show (for k = 1, 2):
$$d\omega(X_0, ... , X_k) = \sum_i (-1)^i X_i(\omega(X_0, ... , \hat{X_i}, ... , X_k)) + \sum_{i < j}(-1)^{i+j}\omega([X_i, X_j], X_0, ... , \hat{X_i}, ... , \hat{X_j}, ..., X_k)$$
In the lecture I'm following we did not learn about Cartans magic formula. I don't even know what $(\mathcal{L}_Y \omega)$ means.
My main problem is: I'm struggling to understand what $X_i(\omega(X_0, ... , \hat{X_i}, ... , X_k))$ is: I thought $\omega(X_0, ... , \hat{X_i}, ... , X_k)$ is an element of $C^{\infty}(U)$. How can we feed it to $X_i$?