Jacobian Matrix of a Local Smooth Extension on an Admissible Domain

Let $n$ and $m$ be natural numbers, let $\Omega$ be an admissible domain in Euclidean space $\mathbb{R}^n$ in the sense of Continuous n-Form, Support, and Zero Extension on a Euclidean or Half-Space Domain, and let $x\in\Omega$. For a map $G$ into $\mathbb{R}^m$ and $k\in\{1,\dots,m\}$ write $G_k$ for the $k$th coordinate function of $G$. Then the following hold.

1. Let $W\subseteq\mathbb{R}^n$ be open with $x\in W$, and let $G:W\to\mathbb{R}^m$ be smooth. Then for all $k\in\{1,\dots,m\}$ and $j\in\{1,\dots,n\}$ the partial derivative of $G_k$ with respect to the $j$th variable exists at every point of $W$; consequently the Jacobian matrix $DG(a)$ is defined for every $a\in W$.

2. Let $W$ and $\widetilde{W}$ be open subsets of $\mathbb{R}^n$ with $x\in W$ and $x\in\widetilde{W}$, and let $G:W\to\mathbb{R}^m$ and $\widetilde{G}:\widetilde{W}\to\mathbb{R}^m$ be smooth maps such that $G(y)=\widetilde{G}(y)$ for every $y\in W\cap\widetilde{W}\cap\Omega$. Then

$$DG(x)=D\widetilde{G}(x).$$