What shown below is a reference from the text Analysis on Manifolds by James Munkres
Definition
Let $S$ be a subset of $\Bbb R^k$; let $f:S\rightarrow\Bbb R^n$. We say that $f$ is of class $C^r$ on $S$ if $f$ may be extended to a function $g:U\rightarrow\Bbb R^n$ that is of class $C^r$ on an open set $U$ of $\Bbb R^k$ containing $S$.
Lemma
Let $U$ be open in $H^k$ but not in $\Bbb R^k$; let $\alpha:U\rightarrow\Bbb R^k$ be of class $C^r$. Let $\beta:U'\rightarrow\Bbb R^n$ and $\gamma:U''\rightarrow\Bbb R^n$ be two $C^r$ extensions of $\alpha$ defined on an open subsets $U'$ and $U''$ of $\Bbb R^k$. Then the derivatives of $\beta$ and $\gamma$ are equal in $U$ and in $\overset{\circ} U$ are equal to the derivative of $\alpha$.
Definition
Given $x\in\Bbb R^n$ we define a tangent vector to $\Bbb R^n$ at $x$ to be a pair $(x;\vec v)$ where $\vec v\in\Bbb R^n$. The set of all tangent vectors to $\Bbb R^n$ at $x$ forms a vector space if we define $$ (x;\vec v)+(x;\vec w):=(x;\vec v+\vec w)\\ c(x;\vec v):=(x;c\vec v) $$ It is called the tangent space to $\Bbb R^n$ at $x$ and it is denoted $\mathcal T_x(\Bbb R^n)$.
Definition
Let be $A$ open in $\Bbb R^k$ or in $H^k$; let $\alpha: A\rightarrow\Bbb R^n$ be of class $C^r$. Let $x\in A$, and let $p=\alpha(x)$. We define a linear transformation $$ \alpha_*\mathcal T_x(\Bbb R^k)\rightarrow\mathcal T_p(\Bbb R^n) $$ by the equation $$ \alpha_*(x;\vec v):=\big(p;D\alpha(x)\cdot\vec v\big) $$ and we call it pushforward of $\alpha$.
Definition
Let $M$ be a $k$-manifold of class $C^r$ in $\Bbb R^n$. If $p\in M$ chose a coordinate patch $\alpha:U\rightarrow V$ about $p$, where $U$ is open in $\Bbb R^k$ or $H^k$. Let $x$ be the point of $U$ such that $\alpha(x)=p$. The set of all vectors of the form $\alpha_*(x;\vec v)$ where $\vec v$ is a vector in $\Bbb R^k$ is called the tangent space to $M$ at $p$ and it is denoted $\mathcal T_p(M)$. Said differently $$ \mathcal T_p(M):=\alpha_*\big(\mathcal T_x(\Bbb R^k)\big) $$
Definition
Let $A$ be an open set $\Bbb R^n$. A $k$-tensor field in $A$ is a function $\omega$ assigning to each $x\in A$ a $k$-tensor defined on the vector space $\mathcal T_x(\Bbb R^n)$. That is, $$ \omega(x)\in\mathcal L^k\big(\mathcal T_x(\Bbb R^n)\big) $$ for each $x\in A$. Thus $\omega(x)$ is a function mapping $k$-tuples of tangent vectors to $\Bbb R^n$ at $x$ into $\Bbb R$; as such, its value on a given $k$-tuple can be written in the form $$ \big[\omega(x)\big]\big((x;\vec v_1),...,(x;\vec v_k)\big) $$ We require this function to be continuous as a function of $(x,v_1,..,v_k)$; if it is of class $C^r$ we say that $\omega$ is a tensor field of class $C^r$. If it happens that $\omega(x)$ is an alternanting $k$-tensor for each $x$, then $\omega$ is called a differential form of order $k$ on $A$. More generally, if $M$ is an $m$-manifold in $\Bbb R^n$ then we define a $k$-tensor field on $M$ to be a function $\omega$ assigning to each $p\in M$ an element of $\mathcal L^k\big(\mathcal T_p(M)\big)$. If in fact $\omega(p)$ is alternanting for each $p$ then $\omega$ is called a differential form on $M$.
In particular any tensor field on $M$ can be extended to a tensor field defined on an open set of $\Bbb R^n$ containing $M$ but the proof is decidedly non-trivial.
Definition
Let $A$ be open in $\Bbb R^k$; let $\alpha:A\rightarrow\Bbb R^n$ be of class $C^\infty$; let $B$ be an open set of $\Bbb R^n$ containing $\alpha[A]$. We define the pull-back $\alpha^*$ of $\alpha$ to be a linear transformation $$ \alpha^*:\Omega^l(B)\rightarrow\Omega^l(A) $$ such that if $f$ is a zero form on $B$ then $$ [\alpha^*f](x):=f\big(\alpha(x)\big) $$ for each $x\in A$ and if $\omega$ is a $l$-formon $B$ with $l>0$ then $$ \big[(\alpha^*\omega)(x)\big]\big((x;\vec v_1),...,(x;\vec v_l)\big):=\big[\omega\big(\alpha(x)\big)\big]\big(\alpha_*(x;\vec v_1),...,\alpha_*(x;\vec v_l)\big) $$ for each $x\in A$ and for each $\vec v_1,..,\vec v_k\in\Bbb R^k$.
Now having defined the differential operator $d$ for forms defined over an open set let's go to define it for forms defined over a manifold. So previously we observe that the pushforward of a coordinate patch is a injective function and thus if $\omega$ is a $l$-form defined over a $k$-manifold we define a $(l+1)$-form $d\omega$ by letting that $$ \big[d\omega(y)\big]\big((y;\vec v_1),...,(y;\vec v_l),(y;\vec v_{l+1})\big):=\Big[d(\alpha^*\omega)\big(\alpha^{-1}(y)\big)\Big]\big((\alpha_*)^{-1}(y;\vec v_1),...,(\alpha_*)^{-1}(y;\vec v_l),(\alpha_*)^{-1}(y;\vec v_{l+1})\big) $$ for any $y\in M$ and for any $\vec v_1,..,\vec v_k,\vec v_{l+1}\in\Bbb R^n$ such that $(y;\vec v_1),...,(y;\vec v_l),(y;\vec v_{l+1})\in\mathcal T_y(M)$.
So since the domain of a coordinate patch of an interior point is a open subset of $\Bbb R^k$ the above position is consistent with what is above defined; anyway if $y$ is a boundary poin of $M$ then the domain of any coordinate patch about $y$ is not open in $\Bbb R^k$ so that the pull-back of this cordinate system is not defined and moreover if it was defined apparently it would carry a differential forms defined over a manifold to differetial forms defined over an open subset in $H^k$ that is not open in $\Bbb R^k$ so that the form $d(\alpha^*\omega)$ would not be defined. So could someone help me, please?