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    \begin{center}
    \textbf{ MATH534A, Problem Set 3, due Sep 18}
    \end{center}
    All problems are worth the same number of points.
    \begin{enumerate}
    \item Prove that $\R P^1$ is diffeomorphic to $S^1$.
    \item Let $S^2 = \{(x,y,z) \in \R^3 \,|\, x^2 + y^2 + z^2 = 1\}$ be the unit sphere, and let $p \in S^2$. Let also $D$ be a $1$'st order differential operator on $S^2$ at $p$, i.e. a rule that assigns a real number to any function which is defined and smooth in some neighborhood of $p$ and has the following properties:
    \begin{itemize}
    \item $D(cf) = c D(f)$ for any smooth function $f$ and any $c \in \R$.
    \item $D(f+g) = D(f) + D(g)$ in the intersection of domains of $f$ and $g$.
    \item $D(fg) = f(p)D(g) + D(f)g(p)$.
    \end{itemize}
Show that there exists a vector $v \in \R^3$ tangent to $S^2$ at $p$ such that $D$ is the differentiation in direction $v$.
    \item The above construction defines an isomorphism $i$ between $T_p S^2$ (defined as the space of $1$'st order differential operator on $S^2$ at $p$) and the geometric tangent plane at $p$. Assume that $p$ belongs to the northern hemisphere. Then we have a chart $(x,y)$ around $p$, and the associated basis $\dbyd x,\dbyd y \in T_pM$. Find $i(\dbyd x), i(\dbyd y)$.
    \item Problem 3-8 from the textbook.

    \end{enumerate}
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