This is an exercise from John Lee's book. I am having extreme difficulty with constructing explicit boundary charts for $M$. I have no idea at all how to explicitly express the boundary charts for $M$. Even for $n=2$, the explicit formula for boundary charts seems impossibly difficult... Could anyone please help me??
This exercise is from the first edition of my book. In the second edition, I added a hint that might help you:
Consider the map $\pi\circ\sigma^{-1}\colon \mathbb R^n \to \mathbb R^n$, where $\sigma\colon \mathbb S^n\smallsetminus\{N\}\to \mathbb R^n$ is the stereographic projection and $\pi$ is a projection from $\mathbb R^{n+1}$ to $\mathbb R^n$ that omits some coordinate other than the first.