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I am trying to come up with a diffeomorphism of the upper half plane $y> 0$ onto the first quadrant $x> 0 , y> 0$

Can anyone come up with such a diffeomorphism?

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    I suggest you write it down, show it's differentiable. Then, write down its inverse and show it's differentiable. This is one of those cases where you should just sort of do it. If you encounter trouble, edit your question with your attempt and we can go from there perhaps.2012-03-02

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Why not $(x,y)\mapsto (e^x, y)$?