# Math Help - Help on smooth manifold!

1. ## Help on smooth manifold!

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.

2. ## Re: Help on smooth manifold!

We cannot find a well defined tangent space at t=0.

3. ## Re: Help on smooth manifold!

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 $f: M \supseteq U \rightarrow V \subseteq \mathbb{R}^n \ with \ x \in U$ (which means that for every open subset of M there is a chart (=diffeomorphism) that maps U to an open subset of $\mathbb{R}^n$).