I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question)

If you don't have the book or need information, then please leave a comment and I will try my best.


$\textbf{Theorem:}$ The theorem says that the transformation $(p,q) \mapsto (I,\phi)$ is symplectic, where $I$ are the action variables and $\phi$ the action angles.

He says that he will only sketch the proof which might be the source of confusion. 

I will state the proof up to the point that causes the problems and explain what exactly causes the troubles. 

$\textbf{Proof: }$ So first we consider the $1$-form $pdq$ on the manifold $M_f:=\{(p_1,..,p_n,q_1,..,q_n)=:(p,q) \in M; F_1(p,q)=f_1,...,F_n(p,q)=f_n\}$ where $F_1,..,F_n$ have linearly independent derivatives and $M$ is a symplectic manifold of dimension $2n$. 

addendum: It can be shown that $\omega|_{M_f} = 0$ and he also assumed that $\frac{\partial I}{\partial f}|_{M_f}$ is invertible in a previous proof.


Therefore, $S(x)= \int_{x_0}^{x} pdq|_{M_f}$ is invariant under deformations of paths $(x_0 \rightarrow x)$ (by Stokes' theorem). 

addendum: It can be shown that if $M_f$ is connected and compact it is diffeomorphic to a torus.

Still, $S$ is multiple-valued as when we integrate around one circle $\gamma_i$ of this torus, we get a period $\Delta_i (S)= \int_{\gamma_i} dS = 2 \pi I_i.
$

Now he continues by saying: Let $x_0$ be a point on $M_f$, in a neighbourhood of which the $n$ variables $q$ are coordinates of $M_f$ such that the submanifold $M_f \subset \mathbb{R}^{2n}$ is given by $n$-equations of the form $p= p(I,q)$, $q(x_0)=q.$ 

In a simply connected neighborhood of the point $q_0$ a single-valued function is defined 

$S(I,q) = \int_{q}^{q} p(I,q) dq.$


Finally, he remarks that: It is not difficult to verify that these formulas actually give a canonical transformation, not only in a neighborhood of the point under consideration but also "in the large" in a neighborhood of $M_f$. 

$\textbf{Question:}$ Now my question is: Why is it possible to take $(I,q)$ as coordinates, i.e. what is the argument that explains why the coordinates $q$ can be taken as even global coordinates (as Arnold actually says that it is easy to conclude that this also holds true in the "large " around $M_f$) in a nbh of $M_f$?



EDIT: For those of you who don't have the book, you can download the pdf from this link and go to page 300 (according to the pdf).[click me.][1]


  [1]: https://www.google.de/url?sa=t&rct=j&q=&esrc=s&source=web&cd=4&cad=rja&uact=8&ved=0CEMQFjADahUKEwjM8cqtp_LHAhVCWBQKHRaNCQs&url=https%3A%2F%2Floshijosdelagrange.files.wordpress.com%2F2013%2F04%2Fv-arnold-mathematical-methods-of-classical-mechanics-1989.pdf&usg=AFQjCNGcX9xCKOCoGWmNB4N_c3MM46KfZQ&sig2=c9GZk5lEijFY9wsjypj_QQ