Let $n,k\in\mathbb{N}$ with $1\le k\le n$, let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $\omega$ be a \reftext{def:c1-differential-k-form-euclidean-open-set-2026b}{$C^1$ differential $(k-1)$-form} on $U$, and fix strictly increasing indices

Write $\omega$ in the coordinate expansion from \ref{thm:coordinate-expansion-differential-forms-euclidean-2026b} as

For each $r\in\{1,\dots,k\}$, let $I^{\hat r}$ denote the increasing $(k-1)$-tuple obtained from $(i_1,\dots,i_k)$ by omitting $i_r$, and let $a_{I^{\hat r}}$ be the corresponding coefficient function. Then the coefficient of

in the coordinate expansion of $d\omega$ from \ref{def:exterior-derivative-c1-differential-form-euclidean-open-set-2026b} is