can anyone give me an example of diffential map of a map $f : M \Rightarrow N $ where $M$,$N$ are manifolds different from Euclidean space?I read somewhere that Hopf map is differential map from $S^3$ to $S^2$ but i don't know about it, Can someone explain it with how it is a differential map?
Take a circle and rotate every point by a fixed amount. This is a differentiable map between the manifolds $M=N=S^1$. You can use charts to show the map is differentiable, and you can calculate its Jacobian. Or take any n-sphere $S^n$ and send a point to its antipode, i.e., send $(x_1,x_2,...,x_n) \rightarrow (-x_1,-x_2,...,-x_n)$