Let $d\in\mathbb N$, $k\in\{1,\ldots,d\}$ and $M$ be a $k$-dimensional embedded $C^1$-submanifold of $\mathbb R^d$ with boundary, i.e. for every $x\in M$, there is an open subset $\Omega$ of $M$ with $x\in\Omega$ and a $C^1$-diffeomorphism from $\Omega$ onto an open subset of $\mathbb H^k:=\mathbb R^{k-1}\times[0,\infty)$. $(\Omega,\phi)$ is called a chart for $M$ and we define $$M^\circ:=\{x\in M:\exists\text{chart }(\Omega,\phi):\phi(\Omega)\text{ is }\mathbb R^k\text{-open}\}$$ and $$\partial M:=\{x\in M:\exists\text{chart }(\Omega,\phi):\phi(x)\in\partial\mathbb H^k\}.$$ $M^\circ$ and $\partial M$ are disjoint.
Now let $(\Omega,\phi)$ be any chart for $M$. It's then easy to see that $$\Omega^\circ=\{x\in\Omega:\phi(x)\not\in\partial\mathbb H^k\}\tag1$$ and $$\partial\Omega=\{x\in\Omega:\phi(x)\in\partial\mathbb H^k\}\tag2.$$ Moreover, if $T$ is a $C^1$-diffeomorphism from $\mathbb R^d$ onto $\mathbb R^d$, $\Omega':=T(\Omega)$ and $$\phi':=\left.\phi\circ T^{-1}\right|_{\Omega'},$$ then $(\Omega',\phi')$ is a chart for $\Omega'$, $$(\Omega')^\circ=T(\Omega^\circ)\tag3$$ and $$\partial\Omega'=T(\partial\Omega)\tag4.$$
Are we able to extend this "local" result to all of $M$, i.e. are we able to conclude that $$(M')^\circ=T(M^\circ)\tag5$$ and $$\partial M'=T(\partial M)\tag6,$$ where $M':=T(M)$?