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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.

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    Well, I posted an answer which happened to miss a rather important point. I tried to correct things but somehow I erased both the answer and the comment that brought that tho my attention. Sorry, didn't mean to...in fact, I didn't even know I can erase others' comments!2012-05-23
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    At least like this, the second function gives only the south cap of the sphere, so that it doesn`t have a inverse. It this well written?2012-05-23
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    @JuanSimões: Yes it is well written. I came across with this in a proof that $ S^2$ is a diffential manifold.2012-05-23
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    @JuanSimões: Sorry you are right! I have edit the question.2012-05-23

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