Assume that we have an open subset $P \subset \mathbb{R}^2$ defined by $P =\{ \mathbf{p}: a < ||\mathbf{p}||_{2} < b \}$. $P$ can be covered by a single chart, which is the identity. This chart can be shown to be homeomorphic, smooth and has a smooth inverse; therefore, it is smooth. Since all of $P$ is covered by this one smooth chart, we have a smooth structure on $P$.
My question is, if I wanted to use more than one chart, what choices would I use?
I thought of the exponential map: chart 1: $re^{i\theta}$ where $a < r< b$ and $0 < \theta < 2\pi$ and chart 2: $re^{i\theta}$ where $a < r< b$ and $-\pi < \theta < \pi$, but it does not map to $\mathbb{R}^2$. I'm really confused that complex plane has 2 dimensions and yet the exponential map results in a single value. Which is why I cannot justify using this map for $P$, but can justify it as a chart for a circle. Please advise!