# Thursday, August 20

:::{.exercise}
Show that \( \theset{(\RR^1, \id), (\RR^1, x\mapsto x^3)} \) is *not* a smooth atlas.
:::

:::{.exercise}
Let $S^1\da\ts{(x, y) \in \RR^2 \st x^2 + y^2 = 1}$ with charts given by stereographic projection from $(0, 1)$ and $(0, -1)$ on $U = S^1\sm\ts{(0, 1)}\to \RR$ and $V = S^1\sm\ts{(0, -1)}\to \RR$.

Show that this gives a smooth atlas.
:::

:::{.exercise}
Write down a smooth atlas on the unit square.
:::