let $f : \mathbb{R}^2 \rightarrow \mathbb{R}^{+*}$ be a smooth function.
Prove that $g:\mathbb{S}^1 \rightarrow \mathbb{R}^2: u \rightarrow g(u) = f(u)u$ is an embedding.
I found that $h: \mathbb{R}^2 \rightarrow \mathbb{S}^1: v \rightarrow h(v)=v/f(v)$ is the inverse of f so f is a homeomorphism. but I can't prove that f is an immersion. Could you please help?
Thank you
Choose any $t_0\in [0,2\pi]$. Let $\alpha(t):=(\cos t,\sin t)$. Then it is enough to show that $$\left. \dfrac{d}{dt} g(\alpha(t)) \right|_{t=t_0} \neq 0.$$ But this is clear because $\alpha $ is orthogonal to $\alpha'$, and $$\dfrac{d}{dt}g(\alpha)=\frac{d}{dt}(f(\alpha))\ \alpha+f(\alpha) \ \alpha'.$$