Can someone give me an example of two compatible atlas for the same differential structure? I'm a student of physics and i just began studying this topic. I read the definition of "compatible atlas" but i can't make clear examples of it by myself. Can someone help me?

Here's a simple example. Take the usual atlas for the unit circle consisting of the four charts by (open) semicircles (two projecting to the $x$-axis, two projecting to the $y$-axis). Call these maps $\phi_i$, $i=1,2,3,4$. Now take an atlas consisting of the two charts by stereographic projection from the north pole and south pole respectively. Call these maps $\psi_j$, $j=1,2$. The two atlases are compatible, because on overlaps the maps $\phi_i\circ\psi_j^{-1}$ and $\psi_j\circ\phi_i^{-1}$ are smooth maps from appropriate open subsets of $\Bbb R$ to $\Bbb R$.
