Let $ \displaystyle{ S^2 = \{(x,y,z) \in \mathbb R ^3 : x^2 + y^2 + z^2 =1 \} }$ and $ \displaystyle { U= \{ (x,y) \in \mathbb R ^2 : x^2 + y^2 < 1 \}}$. Consider the functions
$ \displaystyle{ f_1,f_2: U \to S^2 }$ where $ \displaystyle{ f_1 = (u,v, \sqrt{1-u^2 -v^2}) }$ and $ \displaystyle{ f_2 = (u,v, -\sqrt{1-u^2 -v^2}) }$.
Prove that $ f_2 ^{-1} \circ f_1$ is differentiable.
Do I have to find $ f_2 ^{-1} \circ f_1$ or I can prove that is differentiable without finding the function ?
Can you give some help?
Thanks in advance!
Edit: I still haven't made any progress. Any ideas? Thank you!
Sorry I made a mistake. It is $ f(U_1) \cup f(U_2) = S^2 - \{ (x,y,z) \in \mathbb R ^3 : z=0 \} $. So $ f_2 ^{-1} \circ f_1 $ is differentiable on this set.