Q:

Examples of Smooth Maps

Definition: $F$ is a smooth map if for every $p \in M$, there exists smooth charts $(U,\phi)$ containing $p$ and $(V,\psi)$ containing $F(p)$ such that $F(U) \subseteq V$ and the composite map $\psi \circ F \ \circ \phi^{-1}$ is smooth from $\phi(U)$ to $\psi(V)$. 
So, I'm trying to come up with examples of smooth maps on smooth manifolds and I'm running into a small issue. With the map $\psi \circ F \ \circ\phi^{-1} :\phi(U) \rightarrow \psi(V).  $ For example, I tried $F: \mathbb R^{2}  \rightarrow \mathbb R $ with $F(x,y) = d$ to be the constant function, then I'm trying to figure out what $\phi(U)$ and $\psi(V)$ should be. Also, would $\phi$ be a coordinate map from $\mathbb R^{2} \rightarrow \mathbb R$, and how would I similarly define $\psi$. 
I'm finding the definition of smooth map to be a little confusing. My main problem is trying to come up with examples to make sense of how the charts work as well as the composition map $\psi\circ F \ \circ\phi^{-1} :\phi(U) \rightarrow \psi(V) $. 
Any advice and maybe some examples to show how to correctly apply the definition of smooth map . 

A:

For example, let $(a_{i,j})_{i,j = 0}^n \in \operatorname{GL}_{n+1}(\mathbf{R})$ be an $(n + 1)\times(n + 1)$ matrix. Define a map $$ f : \mathbf{RP}^{n} \to\mathbf{RP}^{n} $$
by $$ f([x^0:x^1:\dots:x^n]) = \left[ a_{0,k}x^k : a_{1,k}x^k : \cdots : a_{n,k}x^k \right] $$
where $a_{i,k}x^k$ is short for $\sum_{k = 0}^n a_{i,k}x^k$.
Recall that an atlas for $\mathbf{RP}^n$ is given by the collection of maps
$$ \varphi_i : U_i := \{[x^0:\cdots:x^n] \mid x^i \ne 0 \} \to \mathbf{R}^n $$
defined by
$$ \varphi_i([x^0:\cdots:x^n]) = \left( \frac{x_0}{x_i}, \dots, \widehat{\frac{x_i}{x_i}} , \dots, \frac{x_n}{x_i} \right) $$
where we omit the $i$-th entry.

Look at the maps $\phi_j \circ f \circ \phi_i^{-1}$. Let $p = [x^0 : \dots : x^n] \in U_i$ and suppose wlog that $x^i = 1$. Then
\begin{align}
\varphi_j \circ f \circ \varphi_i^{-1}(x^0,\dots,x^{i-1},x^{i+1},\dots,x_n) &= \varphi_j \circ f(p) \\
&= \varphi_j (\left[ a_{0,k}x^k : a_{1,k}x^k : \cdots : a_{n,k}x^k \right]) \\
&= \left( \frac{a_{0,k}x^k}{a_{j,k}x^k} , \dots, \widehat{\frac{a_{j,k}x^k}{a_{j,k}x^k}}, \dots, \frac{a_{n,k}x^k}{a_{j,k}x^k} \right)
\end{align}
which is defined whenever $a_{j,k}x^k \ne 0$ (which is an open set since the left hand side is continuous).
So notice, that in all of these charts, $f$ is defined by rational functions and is therefore smooth.

