I am trying to show: the annulus defined by $P = \{\mathbf{p}: r_i < ||\mathbf{p}||_{2} < r_o\} \subset \mathbb{R}^2 $ admits a smooth structure.
The gist of my solution: I first show that $P$ is a surface using surface patches/parametrizations. Since the collection of surface patches covers $P$, it forms an atlas. In my case, I use only 2 pacthes, and both patches use the same parametrization. I can show that the parametrization is $C^{\infty}$, by verifying that it has continuous partial derivatives of all orders. Next, I can check that the determinant of the Jacobian of the parametrization is non-zero at all values of $u$ and $v$, for both parametrizing domains. This tells me that the parametrization is a diffeomorphism. To establish that the atlas is smooth, I need to check if the transition function: parametrization 2 $\circ$ parametrization 1$^{-1}$ is a diffeomorphism, and vice-versa. Since a composition of diffeomorphisms is still a diffeomorphism, both transition functions are diffeomorphic. Therefore, the atlas admits a smooth structure.
Parametrization 1: $\text{ for } u = (0, 2\pi) \text{ and } v = (0,1)$ \begin{eqnarray} f_x = [(1-v)r_i + vr_o] cos(u) \\ f_y = [(1-v)r_i + vr_o] sin(u) \\ \end{eqnarray}
Parametrization 2: $\text{ for } u = (-\pi, \pi) \text{ and } v = (0,1)$ \begin{eqnarray} f_x = [(1-v)r_i + vr_o] cos(u) \\ f_y = [(1-v)r_i + vr_o] sin(u) \\ \end{eqnarray}
Is this correct? If I'm incorrect, is it because of my reasoning or the chosen parametrizations?
Thank you.
*Note: I am an absolute beginner and am self-studying. I am using Introduction to Topological Manifolds & Introduction to Smooth Manifolds by John.M.Lee and Elementary Differential Geometry by Andrew Pressley.