Let $n\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, and let $i\in\{1,\dots,n\}$. The $i$th coordinate $1$-form on $U$ is the \reftext{def:differential-k-form-euclidean-open-set-2026a}{differential $1$-form}

on $U$ defined by

for every point $x\in U$ and every vector $v=(v_1,\dots,v_n)\in\mathbb{R}^n$.