A Zorich proposition about local charts of smooth surfaces

[Unconscious]

From Zorich, Mathematical Analysis II, 1st ed., pag.163:

where the referred mapping (12.1) is a map $\varphi:I_k\to U_S(x_0)$, in which:

1. $I_k\subset\mathbb{R}^n$ is the k-dimensional unit cube,
2. $x_0$ is a generic point on the surface $S$ and $U_S(x_0)$ is a neighborhood of $x_0$ of points of $S$.

I think that there is a deep significance in this theorem that I can't see yet, but anyway, looking brutally at the proof:

we can see that up to this point the author uses the inverse function theorem to say that, if we consider only the first k coordinates both in the domain and in the range, $\varphi$ is actually a diffeomorphism between two suitable neighborhoods $U_{ \mathbb {R}^k_t} (0)$ e $U _ {\mathbb {R} ^ k_x} (\varphi (0))$.
Then, the proof continues in this way:

and here I don't understand why the whole $f: U _ {\mathbb {R} ^ n_x} (\varphi (0)) \to U_ {\mathbb {R} ^ n_t} (0)$ ($n$ instead of $k$) should be a diffeomorphism. Indeed, it seems to me that $t ^ {k + 1}, ..., t ^ n$ are always zero, as $x$ varies.

 

[Office_Shredder]

Is your claim that $f^l(x_1,...,x_n)=x^l$ for $l >k$? That seems unlikely (or you would have had something that was diffeomorphic on the whole space to begin with).

EDIT: sorry, I was too hasty. $f^l$ is only a function of $x_1,...,x_k$, so could not possibly be equal to $x_l$ for $l>k$.

To address your other possible confusion (though maybe you get the point and you're just not sure why it matters), the point of the theorem is if you have a diffeomorphism mapping to some lower dimensional manifold of $\mathbb{R}^n$ you can extend it (at least locally) to a diffeomorphism mapping to all of $\mathbb{R}^n$.

It might help to consider an example. $\phi(x,y)=(x,y,\sqrt{1-x^2-y^2})$ is a function from a small square around the origin in $\mathbb{R}^2$ to a 2 dimensional manifold in $\mathbb{R}^3$. All the proof is doing is saying you can turn this into $\phi(x,y,z)=(x,y,z+\sqrt{1-x^2-y2})$ and now it's a map from a small region around the origin in $\mathbb{R}^3$ to a 3 d submanifold of $\mathbb{R}^3$.

 

[Unconscious]

Thank you, clear.