Show $f: \mathbb{R} \rightarrow S^1$ given by $f(p) = (cos(p), sin(p))$ is a smooth map.

Question

I have no idea how to show that $f: \mathbb{R} \rightarrow S^1$ given by $f(p) = (cos(p), sin(p))$ is a smooth map.

The definition of smooth that we have been given is:

A continuous map $f: M \rightarrow N$ is smooth at $p$ if for any chart $\phi: U \rightarrow U'$ and $\psi: V \rightarrow V'$ the map

$\psi \circ f \circ \phi^{-1} : \phi(U \cap f^{-1}(V)) \rightarrow V'$ is smooth at $\phi(p)$.

I have absolutely no idea how to use this definition.

Thanks!

Answer 1

Hint: Well, on the upper (open) semicircle, a good chart is $x \mapsto (x, \sqrt{1-x^2})$ for $-1 < x < 1$; on the lower semicircle use $x \mapsto (x, -\sqrt{1-x^2})$. On the right and left (open) semicircles, use somethig analogous.