From Munkres:
Could someone clarify why there is no dependence on the $x_1,\dots,x_{k-1}$ coordinates? So we know that for a coordinate patch $\alpha$ we have $$ \alpha\colon U\to V, $$ where $x\in U\subset\mathbb R^k$ and $p=\alpha(x)\in V\subset M$. We know that if $p\in\partial M\cap V$, then $x$ which satisfies $\alpha(x)=p$ lies in $\partial\mathbb H^k$. But I don't see where the (absense of the) dependence of the first $k-1$ comes into play. Apparently we have $g(x_1,\dots,x_{k-1},x_k)=g(x_1',\dots,x_{k-1}',x_k)$, for any $(x_1,\dots,x_{k-1}),(x_1',\dots,x_{k-1}')\in W_0$, but I don't know why.
Also, $W_i=V\cap V'$, where $\beta:U'\to V'$, and $W_0=U\cap U'$. Any hint would be appreciated.