Show that this spiral is a submanifold

Question

I am asked to prove that the set $$ N = \lbrace{(t \cos(1/t), t \sin(1/t)) \in \mathbb{R}^2 | t>0 \rbrace}$$ is a submanifold of $\mathbb{R}^2$. My approach is to show that there exists a chart $(\mathbb{R}^2,h)$ of $\mathbb{R}^2$ for all $p \in N$ s.t. $$ h(N \cap \mathbb{R}^2) = (\mathbb{R}^1 \times \lbrace{0 \rbrace}) \cap h(\mathbb{R}^2).$$ My idea was to use that $g \colon \mathbb{R}^2 \to \mathbb{R}^2,~~(x,y) \mapsto \left( x \cos\left(\frac{1}{x^2+y^2}\right),-y\sin\left(\frac{1}{x^2+y^2}\right)\right)$ is a homeomorphism that fulfills $g(N) = \mathbb{R}_{>0} \times \lbrace{0 \rbrace}$ and then use that as a chart map. With this however I get $$ g(N \cap \mathbb{R}^2) = \mathbb{R}_{>0} \times \lbrace{0 \rbrace} \neq (\mathbb{R}^1 \times \lbrace{0 \rbrace}) \cap g(\mathbb{R}^2)$$ since $g(\mathbb{R}^2) = \mathbb{R}^2$. Can anyone give me a hint as to how I can fix this or tell me if my approach is wrong?

Answer 1

For $x \gt 0$ and in a small enough open subset $W$ of $(x,0)$ the map defined by

$$g(u,v) = ((u+v)\cos 1/ u, (u+v) \sin 1/u)$$ is a $\mathcal C^1$ diffeomorphism of $W \subseteq \mathbb R^2$ onto $U=g(W)$. Moreover we have $g (W \cap ( \mathbb R \times \{0\})) = N \cap U$.

This proves that $N$ is a submanifold of dimension $1$.