I am confused about the procedure for finding the transition functions given an atlas. I understand the theory; it's applying it to real life examples where I have my problem. So for example, take S(adsbygoogle = window.adsbygoogle || []).push({}); ^{1}(the circle). I want to use 2 charts given by:

U_{1}= {α: 0 < α < 2π} φ_{1}= (cos α, sin α)

U_{2}= {β: -π < β < π} φ_{2}= (cos β, sin β)

Now I want to derive the transition function which is where I'm stuck. I know that α = arctan(y/x) and that β = arctan(y/x) which to me implies (and rightly so I think) that α = β on the overlap. My question is what is the transition function?

Another question I have is how are things improved using these 2 patches over 1 patch (which I know fails due to continuity of φ^{-1})?

Thanks in advance for any insight!

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# I S^1 transition functions

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