How do I show that the overlap maps for the real projective space are smooth?

Question

In Lee's Manifolds and Differential Geometry, exercise 1.42 says "Show that the overlap maps for $\mathbb{R}P^n$ are indeed smooth." I'm not sure where to begin. I have read through the text up to this point and feel like I have either missed a particular criteria for a map to be smooth, or I am not connecting the ideas in the necessary way. Any hints to get me moving in the right direction would be greatly appreciated.

If $(U_\alpha, \text{x}_\alpha)$ and $(U_\beta, \text{x}_\beta)$ are charts, then the overlap maps are of form $\text{x}_\beta \circ \text{x}_\alpha^{-1}:\text{x}_\alpha(U_\alpha\cap U_\beta)\rightarrow\text{x}_\beta(U_\alpha\cap U_\beta)$.

Answer 1

For all $i\in\{1,\ldots,n\}$, let define the following open set of $\mathbb{R}P^n$: $$U_i:=\{[x_0:\cdots:x_n];x_i\neq 0\}.$$ Furthermore, let define $\phi_i\colon U_i\rightarrow\mathbb{R}^n$ by: $$\phi_i([x_0:\cdots:x_n]):=\left(\frac{x_0}{x_i},\ldots,\frac{x_{i-1}}{x_i},\frac{x_{i+1}}{x_i},\ldots\frac{x_n}{x_i}\right).$$ Then, $(U_i,\phi_i)_{i\in\{1,\ldots,n\}}$ is a topological atlas for $\mathbb{R}P^n$.

Let us see that the changes of coordinates are smooth, for all $(i,j)\in\{1,\ldots,n\}^2$ such that $j

The key observation is that the inverse of $\phi_i$ is given by: $${\phi_i}^{-1}(x_1,\ldots,x_n)=[x_1:\cdots:x_{i-1}:1:x_{i}:\cdots:x_n].$$