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\begin{document}
\noindent{\small
  \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#1617}{2016-17}:
  \hfill\url{http://drorbn.net/?title=1617-257}
}

\begin{center}

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\noindent{\bf Do not turn this page until instructed.}

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\href{http://drorbn.net/?title=1617-257}{\large Math 257 Analysis II}

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{\LARGE Term Test 2}

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University of Toronto, January 17, 2017

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\noindent{\bf Solve 4 of the 6 problems on the other side of this page. }\\
Each problem is worth 25 points.\\You have an hour and fifty minutes to
write this test.

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\noindent{\bf Notes}
\begin{itemize}
\item No outside material other than stationary is allowed.

\item {\bf Neatness counts! Language counts!} The {\em ideal} written
solution to a problem looks like a page from a textbook; neat and clean
and consisting of complete and grammatical sentences. Definitely phrases like
``there exists'' or ``for every'' cannot be skipped. Lectures are mostly
made of spoken words, and so the blackboard part of proofs given
during lectures often omits or shortens key phrases. The ideal written
solution to a problem does not do that.

\item Do not write on this examination form! Only what you write in the examination booklets counts towards your grade.

{\red
\item In red: post-exam additions/notes.

\item Indicate clearly which problems you wish to have marked; otherwise an arbitrary subset of the problems you solved will be used.
}

\end{itemize}

\end{center}

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\begin{center}

\includegraphics[width=3.5in]{CubicalElephants.png}
\linebreak
{\small\it rectangular approximations of an elephant}

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{\bf Good Luck!}

\end{center}

\newpage

\noindent{\bf Solve 4 of the following 6 problems. } Each problem is worth
25 points. You have an hour and fifty minutes. {\bf Neatness counts! Language counts!}

\vfill\noindent{\bf Problem 1. }
\begin{enumerate}
\item Define ``a function $f\colon\bbR^n\to\bbR^m$ is differentiable at a point $a\in\bbR^n$'' and ``the differential $Df(a)$ of a differentiable function $f\colon\bbR^n\to\bbR^m$ at $a\in\bbR^n$''.
\item State and prove the ``chain rule'' for a composition $g\circ f$ of $f\colon \bbR^n\to\bbR^m$ and $g\colon\bbR^m\to\bbR^p$.
\end{enumerate}

\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. } Let $f\colon\bbR^{k+n}\to\bbR^n$ be of class $C^1$; suppose that $f(a)=0$ and $Df(a)$ has rank $n$. Prove that if $c$ is a point of $\bbR^n$ sufficiently close to $0$, then the equation $f(x)=c$ has a solution.

\vfill\noindent{\bf Problem 3. } {\red (I should have asked to also define ``the integral'', to emphasize that solutions like ``$f$ integrable iff $\mu(D(f))=0$'' are wrong).}
\begin{enumerate}
\item Very carefully, define ``a bounded function $f$ is integrable on a rectangle $Q$''. Assume that your reader does not know the words / phrases ``partition of an interval'', ``partition of $Q$'', ``a rectangle in a partition'', ``the volume of a rectangle'', ``lower sum'', ``upper sum'', and whatever remains.
\item Directly from the definitions, prove that constant functions are integrable.
\end{enumerate}

\vfill\noindent{\bf Problem 4. }
\begin{enumerate}
\item Define: ``A set $G\subset\bbR^2$ is of measure 0''.
\item Prove: If $C\subset\bbR$ is compact, the graph $G$ of a continuous function $f\colon C\to\bbR$ is of measure 0 in $\bbR^2$.
\item Prove: The graph of {\em any} continuous function $f\colon\bbR\to\bbR$ is of measure 0 in $\bbR^2$.
\end{enumerate}

\vfill\noindent{\bf Problem 5. } Use the change of variables theorem and Fubini's theorem to compute the two integrals $\int_{\bbR^2}e^{-(x^2+y^2)/2} dxdy$ and $\int_{\bbR}e^{-x^2/2} dx$ (in this order!). You need to be clear about how you use these two theorems, yet you do not need to explain why functions that you encounter along the way are integrable.

\vfill\noindent{\bf Problem 6. } Let $x_1,\ldots,x_k\in\bbR^n${\red\ (note that $k$ may be strictly smaller than $n$ here)}. Prove:
\begin{enumerate}
\item Interchanging two of these vectors does not change the volume of the parallelepiped they span.
\item Adding a multiple of one of those vectors to another one does not change the volume of the parallelepiped they span.
\item Multiplying one of these vectors by a scalar $\lambda\in\bbR$ multiplies the volume of the parallelepiped they span by $|\lambda|$.
\item If these vectors are orthonormal, the volume of the parallelepiped they span is $1$.
\end{enumerate}

\noindent{\small {\bf Tip. } For this problem, you may freely use everything stated in class.}

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\centerline{\bf Good Luck!}

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