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    \textbf{ MATH534A, Problem Set 1, due Aug 30}
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    All problems are worth the same number of points.
    \begin{enumerate}
    \item Problem 1-1 from the textbook (p. 29).
        \item Problem 1-2 from the textbook (p. 30).
        \item For the sphere $$S^n = \{ (x^1, \dots, x^{n+1}) \in \mathbb R^{n+1} \,|\, \sum_{i=1}^{n+1} (x^i)^2 = 1\},$$ let $$U^i_+ = \{(x^1, \dots, x^{n+1}) \in S^n \,|\, x^i > 0\}, \quad U^i_- = \{(x^1, \dots, x^{n+1}) \in S^n \,|\, x^i < 0\}.$$ Define also $\phi^i_\pm \colon U^i_\pm \to \mathbb R^n$ by $$\phi^i_\pm(x^1, \dots, x^{n+1}) = (x^1, \dots, x^{i-1}, x^{i+1}, \dots, x^{n+1}).$$ Compute the transition maps for the charts $(U^i_\pm, \phi^i_\pm)$. Hence show that $S^n$, covered by these charts, is a smooth manifold.
        \item Let $X$ be a $1$-dimensional manifold covered by two charts with domains $U$ and $V$ respectively. Let also $u \colon U \to \mathbb R$ and $v\colon V \to \mathbb R$ be the associated coordinates. Assume that the transition map is $v = e^u$. Find a cover of $X \times X$ by coordinate charts and compute the corresponding transition maps.
        \item Show that the circle $S^1$ cannot be covered by one chart.
    \end{enumerate}
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