(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given S^{1}={(x,y) in R^{2}: x^{2}+y^{2}=1}. Show that S^{1}is a 1-dimensional manifold.

2. Relevant equations

3. The attempt at a solution

Let f_{1}:(-1,1)->S^{1}s.t. f_{1}(x)=(x,(1-x^{2})^{1/2}).

This mapping is a diffeomorphism from (-1,1) onto the top half of the circle S^{1}.

I was trying to write a prove that f_{1}is indeed a diffeomorphism, but I am having trouble showing the onto part.

I argued that f_{1}is onto by defining the inverse map and showing that the inverse map is 1 to 1 and hence f_{1}is onto.

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# Homework Help: Differential Topology: 1-dimensional manifold

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