TITLE: Finding a diffeomorphism between two smooth structures of $\Bbb R$
QUESTION [7 upvotes]: This is taking from Tu's Introduction to Manifolds book. We have defined $\mathbb{R}$ as the real line with the differentiable structure given by the maximal atlas of the chart $(\mathbb{R},\phi=\operatorname{Id}_\mathbb{R}:\mathbb{R}\to\mathbb{R})$ and $\mathbb{R}'$ as the real line with the differentiable structure given by the maximal atlas of the chart $(\mathbb{R},\psi:\mathbb{R}\to\mathbb{R})$ where $\psi(x)=x^{1/3}$.
We are then instructed to show that there is a diffeomorphism between $\mathbb{R}$ and $\mathbb{R}'$, followed by a hint that it's not the identity map because it is not smooth.
My question(s):
1) Why is the identity map not a diffeomorphism/not smooth?
2) Does anyone have a hint/suggestion for an approach to finding the diffeomorphism that they are looking for?
Many thanks in advance!

REPLY [7 votes]: 1) When composed with the given charts, the identity map gives $(\psi \circ \operatorname{id}_\Bbb R \circ\ \phi^{-1})(x) = x^{1/3}$. This map is not smooth at $0$. Hence the identity map is not a diffeomorphism.
2) Think of a bijection that "cancels out" with the cubic root in $\psi$.

REPLY [3 votes]: The key is that a diffeomorphism is a map $f\colon\ \mathbb R_a\longrightarrow \mathbb R_b$ such that $f$ and $f^{-1}$ is smooth, but the definition of what it means for a map between manifolds to be smooth is in general dependent on the differentiable structure in such a way that if $(U_a,\psi_a)$ is the differentiable structure for $\mathbb R_a$ and $(U_b,\psi_b)$ is the differentiable structure for $\mathbb R_b$, then $f$ is called smooth if the function $\psi_b \circ f \circ \psi_a^{-1}$ is smooth.
1) In your case, $(\psi \circ \mathrm{id} \circ \phi^{-1})(x) = x^{1/3}$ is not smooth at 0.
2) Try choosing $f(x) = x^3$ as your diffeomorphism and work out the details.