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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 3}

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University of Toronto, March 14, 2017

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\noindent{\bf Solve 4 of the 5 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.

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

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

\end{itemize}

\end{center}

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\begin{center}

\includegraphics[width=3.5in]{Figure8Klein.png}
\linebreak
{\small\it A Klein bottle with a segment removed}

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{\bf Good Luck!}

\end{center}

\newpage

\noindent{\bf Solve 4 of the following 5 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 State ``the chain rule'' about the differential of the composition of two functions $f\colon\bbR^n\to\bbR^m$ and $g\colon\bbR^m\to\bbR^p$.
\item By appropriately choosing functions $f\colon\bbR\to\bbR^2$ and $g\colon\bbR^2\to\bbR$, find the derivative of the function $h(x)=x^x${\red, for $x>0$}.
\end{enumerate}

\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.}

\noindent{\small {\bf Tip. } In math exams, ``state'' means ``write the statement of, in full''.}

\noindent{\small {\bf Tip. } In math exams, ``find'' means ``find and explain how you found''.}

\vfill\noindent{\bf Problem 2. } Let $M$ be a subset of $\bbR^n$, and let $B$ be the open unit ball in $\bbR^k$. It is given that two functions $\alpha\colon B\to M$ and $\beta\colon B\to M$ are both homeomorphisms, differentiable of class $C^r$, and their differentials are of maximal rank at every point of $B$. Show that the composition $\beta^{-1}\circ\alpha\colon B\to B$ is a $C^r$ diffeomorphism. (You cannot assume that $\beta^{-1}$ is $C^r$ unless you prove it first).

\par\noindent{\small {\bf Tip. } In math exams, ``show'' means ``prove''.}

\vfill\noindent{\bf Problem 3. }
\begin{enumerate}
\item Write a precise definition of ``the pushforward $\phi_\ast\xi$ of a tangent vector $\xi$ via a $C^r$ map $\phi\colon\bbR^n\to\bbR^m$''.
\item Let $\phi\colon\bbR^2\to\bbR^2$ be given by $\phi(u,v)=(u^2-v^2,2uv)$. Compute $\phi_\ast(\xi)$, where $\xi$ is the tangent vector given as the pair $\left(\begin{pmatrix}0\\1\end{pmatrix},\begin{pmatrix}-1\\0\end{pmatrix}\right)$.
\end{enumerate}

\vfill\noindent{\bf Problem 4. } Consider the forms $\omega = xydx+3dy-yz dz$ and $\eta=xdx-yz^2dy+2xdz$ on $\bbR^3_{xyz}$. Verify by direct computations that $d(d\omega)=0$ and that $d(\omega\wedge\eta)=(d\omega)\wedge\eta - \omega\wedge d\eta$.

\vfill\noindent{\bf Problem 5. } Explain in detail how the vector-field operator $\operatorname{curl}$ arises as an instance of the exterior derivative operator $d\colon\Omega^k(\bbR^n)\to\Omega^{k+1}(\bbR^n)$, for some $k$ and $n$.

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\centerline{\bf Good Luck!}

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\end{document}