How do I show that the overlap maps for the real projective space are smooth? 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 $\text{x}_\beta \circ \text{x}_\alpha^{-1}:\text{x}_\alpha(U_\alpha\cap U_\beta)\rightarrow\text{x}_\beta(U_\alpha\cap U_\beta)$.
 A: For all $i\in\{0,\ldots,n\}$, 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, $(U_i,\phi_i)_{i\in\{0,\ldots,n\}}$ is a topological atlas for $\mathbb{R}P^n$.
Let us see that the changes of coordinates are smooth, for all $(i,j)\in\{0,\ldots,n\}^2$ such that $j<i$, one has:
$$\phi_j\circ{\phi_i}^{-1}(x_1,\cdots,x_n)=\left(\frac{x_1}{x_j},\ldots,\frac{x_{i-1}}{x_j},\frac{1}{x_j},\frac{x_{i+1}}{x_j},\ldots,\frac{x_n}{x_j}\right)$$
which is smooth, since its coordinates are rational function. Otherwise, if $i<j$,one has:
$$\phi_j\circ{\phi_i}^{-1}(x_1,\cdots,x_n)=\left(\frac{x_1}{x_{j-1}},\ldots,\frac{x_{i-1}}{x_{j-1}},\frac{1}{x_j},\frac{x_{i+1}}{x_{j-1}},\ldots,\frac{x_n}{x_{j-1}}\right).$$
Whence the result.
The key observation is that the inverse of $\phi_i$ is given by:
$${\phi_i}^{-1}(x_1,\ldots,x_n)=[x_1:\cdots:x_{i-1}:1:x_{i}:\cdots:x_n].$$
