I am studying differential geometry and I stumbled on something that I don't understand. When we have a m- dim differential manifold, with U_i and U_j open subsets of M with their corresponding coordinate function phi. As can be seen in the figure. If I understand it correctly phi_j of a point gives us a coordinate of that point. So what does the inverse of a coordinate function do? And how come the product of phi_i * phi_j = psi takes us between subsets?