Problem
Problem 1.58 Let $M=\left\{(x, y) \in \mathbb{R}^{2}: x^{2}+y^{2}<1\right\}$. Define a $C^{\infty}$ map by
$$
f: M \rightarrow \mathbb{R}^{2}, \quad(x, y) \mapsto\left(\frac{y}{1-x^{2}-y^{2}}, \mathrm{e}^{x^{2}}\right)
$$
(i) Find the set $S$ of points $p$ of $M$ at which $f_{* p}$ is injective.
(ii) Prove that $f(S)$ is an open subset of $\mathbb{R}^{2}$.