\documentclass[12pt]{article}
\usepackage[margin=0.5in,foot=0.1in]{geometry}
\usepackage{amsmath,amssymb,graphicx,txfonts,multicol}
\usepackage[setpagesize=false]{hyperref}
\usepackage[usenames,dvipsnames]{xcolor}
% Following http://tex.stackexchange.com/a/847/22475:
\usepackage[setpagesize=false]{hyperref}\hypersetup{colorlinks,
  linkcolor={blue!50!black},
  citecolor={blue!50!black},
  urlcolor=blue
}

\def\myurl{http://www.math.toronto.edu/~drorbn/}

\def\red{\color{red}}

\def\bbC{{\mathbb C}}
\def\bbF{{\mathbb F}}
\def\bbQ{{\mathbb Q}}
\def\bbR{{\mathbb R}}
\def\bbV{{\mathbb V}}
\def\bbZ{{\mathbb Z}}
\def\calS{{\mathcal S}}
\def\Aut{\operatorname{Aut}}

\def\problempage{{
  \newpage\noindent\parbox{0.5\linewidth}{
    \footnotesize Solve all 5 problems. Write your solutions only where indicated, or write explicitly, ``continued on page $X$''.
    \newline{\bf Neatness counts! Language counts!}
  }
  \vskip 2mm
  }\parshape 4 0in 0.5\linewidth 0in 0.5\linewidth 0in 0.5\linewidth 0in \linewidth}
\def\yoursolution#1{{
  \vskip 2mm \noindent{\bf Your solution of Problem #1.}
  \newpage\noindent\hfill{\bf Your solution of Problem #1, continued.}
}}
\def\scratchright{{
  \newpage\noindent\hfill\parbox{0.35\linewidth}{\bf
    Scratch work --- this page will not be read unless you explicitly request it.
  }
}}
\def\scratchleft{{
  \newpage\noindent\parbox{0.35\linewidth}{\bf
    Scratch work --- this page will not be read unless you explicitly request it.
  }
}}
\def\scratchboth{{\scratchleft\scratchright}}

% Comment out the following line for the Crowdmark version:
\pagestyle{empty} \def\problempage{\vskip 5mm} \def\yoursolution#1{} \def\scratchright{} \def\scratchleft{} \def\scratchboth{}

\begin{document}
\noindent{\small
  \url{http://drorbn.net/2122-257}
  \newline\href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes/}{Classes}:
  \href{\myurl/classes/\#2122}{2021-22}:
}

\vskip 4mm
\noindent{\bf Do not open this notebook until instructed.}
\vskip 4mm
\begin{center}
  \href{http://drorbn.net/2122-257}{\large Math 257 Analysis II}
  \vskip 4mm
  {\LARGE Term Test 3}
  \vskip 4mm
  University of Toronto, March 8, 2022
  \vskip 5mm{\bf Solve all 5 problems on this booklet. }\\
  The problems are of equal weight.\\You have an hour and fifty minutes to
  write this test.
\end{center}

\vskip 9cm

\noindent{\bf Notes}
\begin{itemize}
\item No outside material allowed other than stationery, minimal hydration and snacks, and stuffed animals.
\item Write your solution of each problem on the problem page and on the back of the problem page. If you run out of space you may continue into the scratch pages, but you {\bf must} indicate this on the problem page or else the scratch pages will not be read.
\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 {\red {\bf In Red.} Comments added after the test took place.}
\end{itemize}

\vfill\begin{center} {\bf Good Luck!} \end{center}\vfill

\scratchright

\problempage
\noindent{\bf Problem 1. } In this question, we say that a function $g\colon\bbR^n\to\bbR^n$ preserves one coordinate if there is some $k\in\underbar{n}$ such that $g_k(x_1,\ldots,x_n)=x_k$, where $g_k$ is the $k$th component function of $g$. Prove that if $n\geq 2$ and $f\colon\bbR^n\to\bbR^n$ is continuously differentiable and $f'(0)$ is invertible, then on a neighborhood of $0$ we can write $f=g_1\circ g_2$ where $g_1$ and $g_2$ are continuously differentiable functions $\bbR^n\to\bbR^n$ and each preserves one coordinate.

\vskip 2mm\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.}

\vskip 2mm\noindent{\small {\bf Tip. } You may want to start by writing ``draft solutions'' on the last pages of this notebook and only then write the perfected versions in the space left here for solutions.}

\vskip 2mm\noindent{\small\red {\bf Ouch. } This problem is buggy. During the test I added the necessary assumption $\exists i\,\frac{\partial f_i(0)}{\partial x_i}\neq 0$. Some further comments:
\begin{itemize}
\item The functions $g_1$ and $g_2$ need only be defined on a neighborhood of $0$, and not on all of $\bbR^n$ as was written. I did not make this comment during the test because it seemed minor and clear enough, and no points will be deducted if you completely ignored this issue.
\item The naming $g_1$ and $g_2$ conflicts with the notation for the components of a function, $g_k$. Strictly sppeaking, this is not a bug. It's just a point of inelegance. Sorry.
\item With the new assumption the condition that $f'(0)$ is invertible is no longer needed.
\end{itemize}}

\yoursolution{1}

\problempage
\noindent{\bf Problem 2. } Let $\phi\colon\bbR^2_{x,y}\to\bbR^2_{u,v}$ be given by $\phi(x,y)=(e^x\cos y,e^x\sin y)$. Compute $\phi^\ast(du\wedge dv)$ and $\phi_\ast\xi$, where $\xi$ is the tangent vector to $\bbR^2_{x,y}$ given by $\xi=\left(\begin{pmatrix}0 \\ \pi/2\end{pmatrix},\begin{pmatrix}0 \\ 1\end{pmatrix}\right)$.

\yoursolution{2}

\problempage
\noindent{\bf Problem 3. } Let $V$ be a vector space, let $\phi\colon V\to V\times V$ be given by $\phi(v)=(v,v)$ and let $\psi\colon V\times V\to V\times V$ be given by $\psi(v,w)=(w,v)$. Let $B\colon V\times V\to\bbR$ be a bilinear function. Prove that $\phi^\ast B=0$ iff $B+\psi^\ast B=0$.

\yoursolution{3}

%\noindent{\bf Problem 3. } Recall that $\bbR^n$, and therefore every tangent space of $\bbR^n$, has a ``standard'' orientation given by the standard ordering of the standard basis $(e_1,\dots,e_n$. We denote this standard orientation We say that a continuously differentiable map $f\colon\bbR^n\to\bb

%\noindent{\bf Problem 3. } We say that a $k$-tensor $\lambda$ on a vector space $V$ is {\em symmetric} if for every permutation $\sigma\in S_k$ we have that $\lambda=\lambda\circ\sigma^\ast$ (namely, if for every $u_i\in V$, we have $\lambda(u_1,\ldots,u_k)=\lambda(u_{\sigma 1},\ldots,u_{\sigma k})$). Let $\calS^kV$ be the vector space of all symmetric $k$-tensors on $V$ (you don't need to show that $\calS^kV$ is a vector space). Assume that $V$ is $n$-dimensional, that $v_1,\dots,v_n$ is a basis of $V$, and that $\varphi_1,\dots,\varphi_n$ is its dual basis. Let $\underbar{n}^k_{nd}$ be the set of sequences $I=(1\leq i_1\leq i_2\leq\dots\leq i_k\leq n)$.
%\begin{enumerate}
%\item Given a sequence $I\in\underbar{n}^k_{nd}$, show how to use $\varphi_{i_1},\ldots,\varphi$
%\end{enumerate}

\problempage
\noindent{\bf Problem 4. } Explain in detail how the vector field operator $\operatorname{curl}$ arises as an instance of the exterior derivative operator $d\colon\Omega^k({\mathbb R}^n)\to\Omega^{k+1}({\mathbb R}^n)$, for some $k$ and $n$.

\vskip 2mm
\noindent{\bf Reminder.} $\def\arraystretch{1.4} % \arraycolsep=1.4pt
\operatorname{curl}\begin{pmatrix}F_1 \\ F_2 \\ F_3\end{pmatrix} = \begin{pmatrix} \frac{\partial{F_3}}{\partial x_2} - \frac{\partial{F_2}}{\partial x_3} \\ \frac{\partial{F_1}}{\partial x_3} - \frac{\partial{F_3}}{\partial x_1} \\ \frac{\partial{F_2}}{\partial x_1} - \frac{\partial{F_1}}{\partial x_2} \end{pmatrix}$.

\yoursolution{4}

\problempage
\noindent{\bf Problem 5. } If $L\colon V\to W$ is an invertible linear transformation between oriented vector spaces (vector spaces equipped with an orientation), we say that $L$ is {\em orientation preserving} if it pushes the orientation of $V$ forward to the orientation of $W$ (or equivalently, if it pulls the orientation of $W$ back to the orientation of $V$). Otherwise, $L$ is called {\em orientation reversing}. Decide for each of the cases below, if $L_i$ is orientation preserving or reversing. In this question ${\mathbb R}^n$ always comes equipped with its standard orientation $(e_1,e_2,\dots,e_n)$.

\begin{enumerate}

\item $L_1\colon{\mathbb R}^2\to{\mathbb R}^2$ via $(x,y)\mapsto(-x,y)$.

\item $L_2\colon{\mathbb R}^2\to{\mathbb R}^2$ via $(x,y)\mapsto(y,x)$.

\item $L_3\colon{\mathbb R}^2\to{\mathbb R}^2$, the counterclockwise rotation by $2\pi/7$.

\item $L_4\colon{\mathbb R}^2\to{\mathbb R}^2$, the clockwise rotation by $2\pi/7$.

\item $L_5\colon{\mathbb R}^2\to{\mathbb R}^2$, the complex conjugation map $z\mapsto\bar{z}$, where ${\mathbb R}^2$ is identified with ${\mathbb C}$ via $(x,y)\leftrightarrow x+iy$.

\item $L_6\colon{\mathbb R}^3\to{\mathbb R}^3$ via $(x,y,z)\mapsto(y,z,x)$.

\item $L_7\colon{\mathbb R}^n\to{\mathbb R}^n$ via $v\mapsto-v$.

\item $L_8\colon{\mathbb R}^m\times{\mathbb R}^n\to{\mathbb R}^n\times{\mathbb R}^m$ via $(u,v)\mapsto(v,u)$, where $u\in{\mathbb R}^m$ and $v\in{\mathbb R}^n$.

\end{enumerate}

\vskip 2mm\noindent{\small {\bf Tip. } The answers for $L_7$ and for $L_8$ may depend on $n$ and $m$.}

\yoursolution{5}

\scratchboth
\scratchboth
\scratchboth
\scratchboth
\scratchboth

\end{document}

