Nonexistence of a global coordinate system I asked this question in https://mathoverflow.net/, but was advised to ask it here. So here it is.  I just started a self-study of differential geometry and topology. And in several text I came accross the question, asking to show that the global coordinates cannot be defined on a circle $S_{1}$. It seems like quite easy question, but I cannot work the proof. I have a hunch that it has something to do with a Jacobian being zero at some points. 
Suppose $S_{1}= [(x,y)\in R^{2} | x^{2}+y^{2}=1] $ and there exist a global coordinate system $u=f(x,y)$, $x=g(x,y)$. The Jacobian then is as follows: 
$$  J=\begin{vmatrix}
\frac{\partial f}{\partial x}  & \frac{\partial f}{\partial y}  \\\ 
\frac{\partial g}{\partial x}  & \frac{\partial g}{\partial y} 
\end{vmatrix}$$
Since $y=\pm \sqrt{1-x^{2}}$ $$\frac{\partial f}{\partial y}=0$$ $$\frac{\partial g}{\partial y}=0$$
So I have that Jacobian is equal to zero. But this also feels not quite right. I'm I missing something?
Comment: the global coordinates means that the mapping is smooth bijective and has a non-zero Jacobian everywhere.
 A: If there exists global coordinates on $S^1$, $S^1$ is homeomorphic to $R^n$ for some $n \ge 1$. But $S^1$ is compact while $R^n$ is not.
A: The notion of "global coordinate system" is somewhat soft. As $S^1$ is a one-dimensional manifold, any coordinate system, local or global, on $S^1$ has one coordinate variable, and not two, as you suggest using $(x,y)$, or $(u,v)$. 
I'm interpreting your question  in the following way: Prove that there is no map
$f:\ ]a,b[\ \to S^1$ which is a diffeomorphism. As $S^1$ is locally diffeomorphically parametrized by the polar angle $\phi$, such a map, if it existed, would have the form
$$f:\quad ]a,b[\ \to S^1,\quad t\mapsto e^{i\phi(t)}\ ,$$
 and would install on $S^1$ a global coordinate system, namely the coordinate $t$ ranging in the interval $\ ]a,b[\ $  of ${\mathbb R}$. 
Assume that $f$ has the required properties. In particular, the function $\phi(\cdot)$ is continuous and injective, therefore monotone, say, monotonically increasing. It follows that $$\alpha:=\lim_{t\to a+}\phi(t)=\inf_{t\in\ ]a,b[}\phi(t)$$ and $$\beta:=\lim_{t\to b-} \phi(t)=\sup_{t\in\ ]a,b[}\phi(t)$$ exist, and
$\alpha<\beta\leq\alpha+2\pi$; otherwise  $f$ would not be injective. But from $\beta\leq \alpha+2\pi$ and the fact that $\ ]a,b[\ $ has neither a minimal nor a maximal element it follows that the value $\alpha+2k\pi$ is not taken by $\phi$ for any $k\in{\mathbb Z}$, whence $e^{i\alpha}\in S^1$ is not taken by $f$. Therefore $f$ would not be surjective. 
It follows that such an $f$ cannot exist.
