Is this map a diffeomorphism?

Question

Consider the map $f:\mathbb{R}\rightarrow T_{(x_1,x_2)}\mathbb{S}^1$ defined by $f(t)=t(x_2,-x_1)$. Is this map a diffeomorphism? I think it is; I am trying to prove that the tangent bundle of $\mathbb{S^1}$ is diffeomorphic to $\mathbb{S}^1\times\mathbb{R}$. I have already proved that it is a homeomorphism and it is smooth; how do I get smoothness of $f^{-1}$? Thank you in advance!

Answer 1

That's a diffeomorphism from the real number line to one tangent line to the circle, yes. In fact, it's linear and invertible, so certainly a diffeomorphism. But what you need is a diffeomorphism from $S^1 \times \mathbb R$ to $TS^1$; when you write that down, we can tell you how to invert it and prove the inverse smooth.