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%\noindent Name: $\underline{\hspace{2.5in}}$ \hfill Student ID: $\underline{\hspace{1.5in}}$

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\begin{center}
  {\Large UNIVERSITY OF TORONTO}\\
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  {\large Faculty of Arts and Sciences}\\
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  {\Large APRIL EXAMINATIONS 2017}\\
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  {\Large Math 257Y1 Analysis II --- Final Exam}

  \vskip 2mm

  Dror Bar-Natan\par
  April 20, 2017\par
\end{center}

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{\bf Solve 6 of the following 8 questions.} If you solve more than 6 questions indicate very clearly on the examination booklet (not on this form) which are the ones that you want marked, or else arbitrary ones may be excluded. The questions are of equal value of 20 points each, even though they might not be of equal difficulty. [In the computations for the final mark, the results of this exam will be scaled back by a factor of $100/120$].

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\noindent{\bf Duration. } You have 3 hours to write this exam.

\vskip 2mm

\noindent{\bf Allowed Material. } None.

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\begin{center}
  \includegraphics[width=4in]{PlanimeterFromWikipedia.png}
  \newline
  Wikipedia: Planimeter: A planimeter (1908) measuring the indicated area by tracing its perimeter.
\end{center}

\vfil

\centerline{{\bf \large Good Luck!}}

\newpage

\noindent{\bf Solve 6 of the following 8 problems. } Each problem is worth
20 points. You have three hours.

\vskip 2mm\par\noindent{\small {\bf Tip. } Neatness, cleanliness, and organization count, here and everywhere else!}

\vfill\noindent{\bf Problem 1.} Let $(X,d)$ be a metric space, let $p_1$, $p_2$, and $p_3$ be points in $X$, and let $U_1$, $U_2$, and $U_3$ be open sets in $\bbR$. Prove that the set
\[ A\coloneqq \left\{x\in X\colon\quad
  d(x,p_1)\in U_1\text{ or\ \ }\biggl( d(x,p_2)\in U_2\text{ and }d(x,p_3)\in U_3\biggr)
  \right\}
\]
is open in $X$.

\vskip 2mm\par\noindent{\small {\bf Tip. } Don't start working! Read the whole exam first. You may wish to start with the questions that are easiest for you.}

\vfill\noindent{\bf Problem 2.} Denote by $B_r$ the open ball of radius $r$ around $0\in\bbR^n$ (using, here and below, the Euclidean metric). It is given that a function $f\colon B_1\to\bbR^n$ satisfies $f(0)=0$ and
\[ \forall x\neq y\in B_1,\quad |f(y)-f(x)-(y-x)|< 0.1|y-x|. \]
\begin{enumerate}
\item Prove that $f$ is one to one.
\item Prove that $f$ is onto $B_{0.4}$. Namely, for every $z\in B_{0.4}$ there is some $x\in B_1$ such that $z=f(x)$.
\end{enumerate}

\vfill\noindent{\bf Problem 3.} Let $Q$ be the rectangle $[a,b]\times[c,d]$ and let $f$ be a real-valued function on $Q$.
\begin{enumerate}
\item Precisely state Fubini's theorem about expressing $\displaystyle\int_Qf$ as an iterated integral.
\item Give an example of a function $g\colon Q\to\bbR$ for which $\displaystyle\int_{\{x\}\times[c,d]}g$ exists for every $x\in[a,b]$, yet for which $\displaystyle\int_Qg$ does not exist.
\end{enumerate}

\vfill\noindent{\bf Problem 4.} Compute the surface area of the two-dimensional sphere $S^2=\left\{x\in\bbR^3\colon |x|=1\right\}$ in $\bbR^3$ by writing it as an integral and then evaluating that integral.

\newpage\noindent{\bf Problem 5.}
\begin{enumerate}
\item Explain how the cross product $a\times b$ of vectors $a,b\in\bbR^3$ can be expressed using $\Lambda^1(\bbR^3)$, $\Lambda^2(\bbR^3)$, and the operation $\wedge$.
\item Define the operations $\operatorname{grad}$, $\operatorname{curl}$, and $\operatorname{div}$, which act on functions or vector fields on $\bbR^3$, and output functions or vector fields on $\bbR^3$.
\item Explain how $\operatorname{grad}$, $\operatorname{curl}$, and $\operatorname{div}$ can be expressed using the ``de-Rham complex''
    \[ \Omega^0(\bbR^3) \stackrel{d}{\longrightarrow} \Omega^1(\bbR^3) \stackrel{d}{\longrightarrow} \Omega^2(\bbR^3) \stackrel{d}{\longrightarrow} \Omega^3(\bbR^3). \]
\end{enumerate}

\vfill\noindent{\bf Problem 6.} Let $\displaystyle\omega=\frac{xdy-ydx}{x^2+y^2}\in\Omega^1\left(\bbR_{x,y}^2\setminus\{0\}\right)$, and let $f\colon Q=(0,\infty)_r\times[0,2\pi]_\theta\to\bbR^2$ be given by $f(r,\theta)=(r\cos\theta,r\sin\theta)$.
\begin{enumerate}
\item Compute $f^\ast(\omega)$.
\item Show that $\omega$ is closed.
\item Show that $f^\ast(\omega)$ is exact on $Q$.
\item Show that $\omega$ is not exact on $\bbR_{x,y}^2\setminus\{0\}$.
\end{enumerate}

\vfill\noindent{\bf Problem 7.} Write a 1-2 paragraph summary description for each of the (approximately 5) significant ingredients appearing in the formula $\displaystyle\int_Md\omega = \int_{\partial M}\omega$. [You are not required to prove this formula, and it is left to you to choose the significant ingredients that should be described].

\vskip 2mm\par\noindent{\small {\bf Tip. } In this problem neatness, cleanliness, and organization especially count, so you may want to write the solution twice; first as a draft, and then in final form. Your summary descriptions must capture the essence of the objects being described, yet they must not be longer than 1-2 paragraph each, or else they will not be read.

\vfill\noindent{\bf Problem 8.} Prove the formula $\displaystyle\int_Md\omega = \int_{\partial M}\omega$ by first regarding forms supported within ``inner'' coordinate charts, then regarding forms supported within ``boundary'' charts, and then globalizing using a partition of unity. [You may assume without proof that orientation details ``work out''].

\vskip 2mm\par\noindent{\small {\bf Tip. } In math exams when proving a major theorem, you may assume as known all material that clearly preceded that proof.

\vfill \label{goodluck}\centerline{\bf \large Good Luck!}\vfill

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