Exterior Derivative of a $C^1$ Differential Form on a Euclidean Open Set

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 $C^1$ differential $k$-form on $U$. Write

$$\omega=\sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\dots i_k}\, dx_{i_1}\wedge\cdots\wedge dx_{i_k}$$

as in Coordinate Expansion of Differential Forms on Euclidean Open Sets. The exterior derivative of $\omega$ is the differential $(k+1)$-form

$$d\omega$$

on $U$ defined by

$$d\omega=\sum_{1\le i_1<\cdots<i_k\le n}\;\sum_{j=1}^n \frac{\partial a_{i_1\dots i_k}}{\partial x_j}\, dx_j\wedge dx_{i_1}\wedge\cdots\wedge dx_{i_k},$$

where the partial derivatives are in the sense of Partial Derivative of a Coordinate Function. When $k=0$, this is the usual differential of a $C^1$ real-valued function.