Let $n\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $k\in\mathbb{N}\cup\{0\}$, and let $\omega$ be a \reftext{def:c1-differential-k-form-euclidean-open-set-2026b}{$C^1$ differential $k$-form} on $U$. Write

as in \ref{thm:coordinate-expansion-differential-forms-euclidean-2026b}. The exterior derivative of $\omega$ is the differential $(k+1)$-form

on $U$ defined by

where the partial derivatives are in the sense of \ref{def:partial-derivative-coordinate-map-2026a}. When $k=0$, this is the usual differential of a $C^1$ real-valued function.