Manifolds Part 2

Problem 1.57 Prove that the $C^{\infty}$ map $$ \Psi: \mathbb{R} \rightarrow \mathbb{R}^{2}, \quad t \mapsto(x, y)=\left(t^{2}, t^{3}\right) $$ (see Fig. 1.18) is not an immersion.

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 $\mathbb{R}^{2}$.

Problem 1.59 Let $\mathbb{R}_{\mathrm{id}}$ and $\mathbb{R}_{\varphi}$ be the $C^{\infty}$ manifolds defined, respectively, by the differentiable structures obtained from the atlases $\{(\mathbb{R}, \mathrm{id})\}$ and $\{(\mathbb{R}, \varphi)\}$ on $\mathbb{R}$, where $\varphi: \mathbb{R} \rightarrow \mathbb{R}, \varphi(t)=t^{3}$. Prove that $\mathbb{R}_{\mathrm{id}}$ and $\mathbb{R}_{\varphi}$ are diffeomorphic (see Problem 1.22).