Active Coordinate Coefficient Formula for the Exterior Derivative

Let $n,k\in\mathbb{N}$ with $1\le k\le n$, let $U\subseteq \mathbb{R}^n$ be open, let $\omega$ be a $C^1$ differential $(k-1)$-form on $U$, and fix strictly increasing indices

$$1\le i_1<\cdots<i_k\le n.$$

Write $\omega$ in the coordinate expansion from Coordinate Expansion of Differential Forms on Euclidean Open Sets as

$$\omega=\sum_{1\le j_1<\cdots<j_{k-1}\le n} a_{j_1\dots j_{k-1}}\,dx_{j_1}\wedge\cdots\wedge dx_{j_{k-1}}.$$

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

$$dx_{i_1}\wedge\cdots\wedge dx_{i_k}$$

in the coordinate expansion of $d\omega$ from Exterior Derivative of a C^1 Differential Form on a Euclidean Open Set is

$$\sum_{r=1}^k (-1)^{r-1}\frac{\partial a_{I^{\hat r}}}{\partial x_{i_r}}.$$