How can I show $(\pi_1,W_1), (\pi_2,W_2)$ is a smooth atlas for $S^3$ Let $S^3 = \{(x^1,x^2,x^3,x^4) \in \mathbb{R}^4 \mid (x^1)^2+(x^2)^2+(x^3)^2+(x^4)^2 = 1\}$
Let $W_1 = S^3 - (0,0,0,1);$ and $\pi_1: W_1\to \mathbb{R}^3$ by 
$$\pi_1(x^1,x^2,x^3,x^4) = \left (\frac{x^1}{1-x^4},\frac{x^2}{1-x^4},\frac{x^3}{1-x^4} \right ) $$
Let $W_2 = S^3 - (0,0,0,-1);$ and $\pi_2: W_2\to \mathbb{R}^3$ by 
$$\pi_2(x^1,x^2,x^3,x^4) = \left (\frac{x^1}{1+x^4},\frac{x^2}{1+x^4},\frac{x^3}{1+x^4} \right ) $$
How can I show that $(\pi_1,W_1), (\pi_2,W_2)$ is a smooth atlas for $S^3$?
I know that in order to show this, I need to show that $\pi_1,\pi_2$ are diffeomorphisms, and that $W_1 \cup W_2 \supseteq S^3$ and $\pi_2 \circ \pi_{1}^{-1}$ is $C^\infty$. 
but I don't know how to show any of these things computationally, how could I do that? 
 A: General directions:


*

*Since $(0,0,0,1)\in W_2$ and $(0,0,0,-1)\in W_1$, it follows that  $\Bbb S^3 = W_1\cup W_2$ by definition of $W_1$ and $W_2$.

*Differentiability of $\pi_1$ and $\pi_2$ only makes sense after we've put a differentiable structure on $\Bbb S^3$. In particular, charts will automatically be diffeomorphisms. That being said, you only have to check that $\pi_1$ and $\pi_2$ are homeomorphisms and that $\pi_2\circ\pi_1^{-1}$ is $C^\infty$: with this, the differentiable structure given by the maximal atlas containing  $\pi_1$ and $\pi_2$ will make them diffeomorphisms.

*To check that $\pi_1$ and $\pi_2$ are homeomorphisms, you can give their inverses explicitly, and see that they're continuous. Continuity of $\pi_1$ and $\pi_2$ is obvious, by their components. 

*For example, you can recognize $\pi_1$ as stereographic projection through the north pole, and $\pi_2$ through the south pole. If $(u^1,u^2,u^3)\in \Bbb R^3$, let $$X(t)= (u^1,u^2,u^3,0)+t((0,0,0,1)-(u^1,u^2,u^3,0)).$$There is an unique $t_0$ such that $X(t_0)\in \Bbb S^3$ (in other words, such that $\|X(t_0)\|=1$). Then $\pi_1^{-1}(u^1,u^2,u^3)= X(t_0)$. Similarly for $\pi_2$.

*Use the above item to give an expression for $\pi_2\circ \pi_1^{-1}(u^1,u^2,u^3)$, and recognize it as $C^\infty$.
