$C^1$ Differential k-Form on an Open Subset of Euclidean Space

Let $n\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be open, let $k\in\mathbb{N}\cup\{0\}$, and let $\omega$ be a differential $k$-form on $U$. Write $\omega$ in the coordinate expansion from Coordinate Expansion of Differential Forms on Euclidean Open Sets. We say that $\omega$ is of class $C^1$ if each coefficient function in that expansion is a $C^1$ map from $U$ to $\mathbb{R}$.