Coordinate 1-Form on an Open Subset of Euclidean Space

Let $n\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be open, and let $i\in\{1,\dots,n\}$. The $i$th coordinate $1$-form on $U$ is the differential $1$-form

$$dx_i$$

on $U$ defined by

$$(dx_i)_x(v)=v_i$$

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