proving that this function is a diffeomorphism i have a question on section 2-3 
I have a question in the book do Carmo Differential Geometry, Section 2-3, Proposition 1, the book says is a composition of $$
h\left| N \right. = F^{ - 1} \left( {y\left| N \right.} \right)
$$
 differential maps, then by the chain rule h is differentiable at r. Since r is arbitrary, h is differentiable on $$
y^{ - 1} \left( W \right)
$$
Ok here I agree on everything, except the last, and I see that this implies that h is a local diffeomorphism, that thing I'm not seeing?
 A: Having just obtained a copy of the book, you are leaving out very key things about the proposition and its proof.  In particular, you do not yet have abstract smooth manifolds, and you have no definition of smooth map from one manifold to another.  At a higher level, the proposition you refer to is essentially a definition, but here you only have the notion of smooth maps from $\mathbb{R}^m$ to $\mathbb{R}^n$.  
Here is the idea of the proof from the book, with some details removed so that calculations don't obscure the ideas.
The problem is that $x^{-1}$ is only defined on (part of) the surface, and not on (a neighborhood of) $\mathbb{R}^3$, and so you can't directly use the chain rule or other basic calculus facts.  Instead, you have to fatten up $x$ to a new map $\widetilde{x}:U\times \mathbb{R}\to S\times \mathbb{R} \to \mathbb{R}^3$.  In slightly fancy language, you change perspectives from looking at the surface $S$ to looking at a tubular neighborhood of $S$.  Because $\widetilde{x}:U\to V$ where $U$ and $V$ are open subsets of $\mathbb{R}^3$, and $\widetilde{x}$ is invertible, we can use the chain rule and the inverse function theorem to conclude that $x^{-1}\circ y = p_{12}\circ \widetilde{x}^{-1}\circ y$ is a local diffeomorphism and a homeomorphism between two open sets in $\mathbb{R}^2$ (here $p_{12}:\mathbb{R}^3\to \mathbb{R}^2$ is the projection onto the first two coordinates).  
Finally, the inverse function theorem implies that a map which is both a homeomorphism and a local diffeomorphism between open subsets of $\mathbb{R}^n$ must be a diffeomorphism.
