There is only one differentiable structure permitted in R^2, meaning, I think, that all atlases in R^2 are diffeomorphic to the Cartesian atlas. But, doesn't the polar coordinate system represent an atlas that is not truly diffeomorphic to the Cartesian atlas, due to the coordinate singularity it has at its origin? 



differentiable structure and coordinates in R^2
I think Matt Noonan's comment technically answered my question. The polar coordinate chart is not a valid atlas on $\mathbb{R}^2$ because the angular coordinate is discontinuous across the non-negative x-axis.


