Why f: R -> R^2 by f(t) = (t^2, t^3) is not a smooth manifold? Just learned the concept of smooth manifold. Really confused about it.
It's because the inverse function is not differentiable at the point x = 0.
A smooth manifold M is a topological space where for every x in M there is a diffeomorphism $\displaystyle f: M \supseteq U \rightarrow V \subseteq \mathbb{R}^n \ with \ x \in U $ (which means that for every open subset of M there is a chart (=diffeomorphism) that maps U to an open subset of $\displaystyle \mathbb{R}^n $).