Hi. I'm thinking about a statement in the example above (do Carmo):
If $\mathbf{x}:U\subset\mathbb{R}^2\to S$ is a parametrization, $\mathbf{x}^{-1}:\mathbf{x}(U)\to\mathbb{R}^2$ is differentiable.
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Being a parametrization, $\mathbf{x}$ is certainly differentiable and has an inverse $\mathbf{x}^{-1}$. Is this strong enough to guarantee that $\mathbf{x}^{-1}$ is differentiable? How could this happen? Thanks.