22.6 Induced atlas on the boundary I am trying to understand one step of Loring Tu's (Intro to Manifolds) proof of Problem 22.6.
Problem

Proof (excerpt)

Question
I don't understand why (*) is not also true for $j = n$, which would mean $\frac{\partial y_n}{\partial x_n} \vert_{\phi^{-1}(a,0)} = 0$. If it were true, the rest of the proof would not go through, so I'm obviously missing something.

If the $n$th partial derivative were also $0$, the upper half-space would map (at least instantaneously) just into the boundary. You would certainly not have a diffeomorphism in that event. Note that ($*$) is derived assuming condition (i), and condition (ii) applies when you're differentiating with respect to $x^n$.
