# Local coordinate systems

Let $\Bbb R$ be the real line with the differentiable structure given by the maximal atlas of the chart ($\Bbb R,f = 1:\Bbb R→\Bbb R$), and let $\Bbb R′$ be the real line with the differentiable structure given by the maximal atlas of the chart $(\Bbb R,y : \Bbb R→\Bbb R)$, where $y(x) = x \frac13$.

(a) Show that these two differentiable structures are distinct.
(b) Show that there is a diffeomorphism between $\Bbb R$ and $\Bbb R′$.

(Hint: The identity map $\Bbb R→\Bbb R$ is not the desired diffeomorphism; in fact, this map is not smooth.)

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 You should probably clarify that $y(x)$ is the cube root of $x$ rather than $x / 3$, and mark it with some tag related to calculus or real analysis, and perhaps homework. – Hew Wolff Nov 28 '12 at 18:46