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    \begin{center}
    \textbf{ MATH534A, Problem Set 4, due Sep 27}
    \end{center}
    All problems are worth the same number of points.
    \begin{enumerate}
    \item Let $\phi \colon M \to N, \psi \colon L \to M$ be smooth maps between smooth manifolds. Use three different definitions of the differential of a smooth map (in terms of velocity vectors of curves, in terms of differential operators, and in terms of the Jacobian matrix) to verify that $d(\phi \circ \psi) = d\phi \circ d\psi$ (i.e. you should present three different proofs).
    \item Let $S^2 \subset \R^3$ be the unit sphere centered at the origin, and let $\phi_\alpha \colon S^2 \to S^2$ be rotation of the sphere by an angle $\alpha$ about the $z$-axis. Compute the differential of $\phi_\alpha$ at the north pole. 
    \item Problem 3-5 from the textbook.
    \item Let $M$ be a smooth manifold. Show that a chart $(U, \phi)$ on $M$ is smooth (i.e. it can be included in a smooth atlas defining the given smooth structure on $M$) if and only if $\phi \colon U \to \phi(U)$ is a diffeomorphism.
    \item (Notation corrected on Sep 20) Let $M$ be a smooth manifold of dimension $n$, and $N$ be its smooth submanifold of dimension\,$m$. By definition, this means that for every $p \in N$ there exists a smooth chart $(U_p, \phi_p)$ in $M$ 
such that $p \in U_p$, and $\phi_p(U_p \cap N) = \phi_p(U_p) \cap \{ x^{m+1} = 0, \dots, x^{n} = 0\}$. Show that the collection $(U_p \cap N, \phi_p\mid_{U_p \cap N})$, where $p \in N$, is a smooth atlas on $N$ which turns $N$ into a smooth manifold of dimension $m$.
    \end{enumerate}
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