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\begin{document}

\def\myurl{http://www.math.toronto.edu/~drorbn}
\noindent{\small \href{\myurl}{Dror Bar-Natan}:
  \href{\myurl/classes}{Classes}:
  \href{\myurl/classes/#2122}{2021-22}:
  \href{https://q.utoronto.ca/courses/237801}{MAT 257 Analysis II}:
}
\hfill{\footnotesize \url{http://drorbn.net/2122-257/ap/Final-Info.pdf}}
  \ \qrcode[height=3em,level=L,nolink]{drorbn.net/2122-257/ap/Final-Info.pdf}

\begin{center}
  {\Large Final Exam Information and Practice Questions}
\end{center}

\begin{itemize}
\input{Final-Details.tex}
\end{itemize}

The 2020/21 Final Assessment:
\begin{enumerate}

\item Let $f\colon{\mathbb R}^n\to{\mathbb R}$ be differentiable at a point $a\in{\mathbb R}^n$, and for $b\in{\mathbb R}^n$ define $L(b):=\lim_{\epsilon\to 0}\frac{f(a+\epsilon b)-f(a)}{\epsilon}$. Prove that $L$ is a linear function of $b$.

\item A function $f\colon{\mathbb R}^n\to{\mathbb R}^n$ satisfies $|(x-y)-(f(x)-f(y))|\leq\frac13|x-y|$ for every $x,y\in{\mathbb R}^n$. Prove that $f$ is continuous.

\item Let $f\colon{\mathbb R}^2\to{\mathbb R}$ be a continuous function whose support is contained in the unit square $[0,1]\times[0,1]$ in ${\mathbb R}^2$, let $s\colon[0,1]\to[0,1]$ be a continuous function, and define $g\colon{\mathbb R}^2\to{\mathbb R}$ by $g(x,y)=f(x,y-s(x))$. Explain why $f$ is integrable on $[0,1]\times[0,1]$ and why $g$ is integrable on $[0,1]\times[0,2]$, and show that
\[ \int_{[0,1]\times[0,1]}f = \int_{[0,1]\times[0,2]}g. \]

{\em Hint.} Make Fubini happy!

\item If $\gamma\colon{\mathbb R}\to{\mathbb R}^n$ is a smooth path and $t\in{\mathbb R}$, let $\dot{\gamma}(t)$ be the tangent vector to ${\mathbb R}^n$ given as the pair $(\gamma(t),\gamma'(t)e_1)$, where $e_1$ is the standard basis vector of ${\mathbb R}$. Show

(a) If $f\colon{\mathbb R}^n\to{\mathbb R}$ is a smooth function, then $D_{\dot{\gamma}(t)}f = (f\circ\gamma)'(t)$, where $D$ denotes the directional derivative.

(b) If $g\colon{\mathbb R}^n\to{\mathbb R}^m$ is smooth and $\eta=g\circ\gamma$, then $\dot{\eta}(t)=g_\ast(\dot{\gamma}(t))$.

\item Let $v_1=\begin{pmatrix}3\\1\end{pmatrix}$ and $v_2=\begin{pmatrix}5\\2\end{pmatrix}$. Together, they form a basis $(v_1,v_2)$ of ${\mathbb R}^2$. Write the dual basis $(\varphi_1,\varphi_2)$ as a pair of row vectors.

\item Let $c_1$ and $c_2$ be singular 1-cubes in ${\mathbb R}^7$, for which $c_1(0)=c_1(1)$ and $c_2(0)=c_2(1)$.

(a) Show that there is a singular 2-cube $c$ in ${\mathbb R}^7$ for which $\partial c=c_1-c_2$.

(b) Suppose now that $c_1(0)=c_1(1)$ but $c_2(0)\neq c_2(1)$. Is it still possible that there is a singular 2-cube $c$ in ${\mathbb R}^7$ for which $\partial c=c_1-c_2$?

\item Suppose a $k$-dimensional manifold $M$ in ${\mathbb R}^n$ is given near a point $p\in M$ as the zero set of a function $z\colon{\mathbb R}^n\to{\mathbb R}^{n-k}$ whose differential is of maximal rank at $p$, and let $\xi\in T_p{\mathbb R}^n$. Show that $\xi\in T_pM$ if and only if $z_\ast\xi=0$. (In doing so, you will have to recall the definition of $T_pM$!)

\item A subset $B$ of ${\mathbb R}^3$ is the union of an infinite line, an infinite ray, and a circle positioned as on the figure below. In addition, oriented loops $R_1$, $R_2$, $G_1$, $G_2$, and $G_3$ are also given as in the same figure. A closed $\omega\in\Omega^1({\mathbb R}^3\setminus B)$ is also given, and it is known that $\int_{R_1}\omega=\pi$ and $\int_{R_2}\omega=e$. Compute $\int_{G_i}\omega$ for $i=1,2,3$.

\[ \includegraphics[width=3in]{../2021-257-AnalysisII/Conf1.png} \]

{\em Hint.} You may want to also think about 2D subsets of ${\mathbb R}^3$ that are shaped like masks, tubes, and/or sacks as in the figure below.

\[ \includegraphics{../2021-257-AnalysisII/MaskTubeSack.png} \]

\end{enumerate}

The following questions were a part of a question pool for the 2020-21 MAT257 Final Assessment, but at the end, they were not included.

\begin{enumerate}

\item Suppose that the bounded functions $f$ and $g$ are integrable over some rectangle $R\subset\bbR^n$. Show that $fg$, $f^2$, and $g^2$ are also integrable over $R$ and that $\int_Rfg\leq\left(\int_Rf^2\right)^{1/2}\left(\int_Rg^2\right)^{1/2}$.

\item Prove that the intersection of finitely many open sets in $\bbR^n$ is open, and give a counterexample to show that this statement may not be true if the intersection is countably infinite.

\item If $A$ and $B$ are disjoint closed sets in $\bbR^n$, show that there exists disjoint open subsets $C$ and $D$ of $\bbR^n$ such that $A\subset C$ and $B\subset D$.

\item Recall that the variation $o(f,t)$ of a function $f\colon\bbR\to\bbR$ at a point $t\in\bbR$ is defined to be
  \[ \lim_{r\to 0}\sup\left\{|f(x)-f(y)|\colon x,y\in B_r(t)\right\}. \]
  Prove that if $f$ is monotone on some interval $[a,b]$ and $P=(a=t_0<t_1<\ldots<t_{n-1}<t_n=b)$ is a partition of $[a,b]$, then
  \[ \sum_{i=1}^{n-1} o(f,t_i) \leq |f(b)-f(a)|. \]

\item Show that the function $f(x,y)=|xy|^{1/2}$ is continuous at $a=(0,0)$ and has both its partial derivatives exist at $a$, yet it is not differentiable at $a$.

\item Prove that if a differentiable function $f\colon\bbR^n\to\bbR$ has a local max at some point $a\in\bbR^n$, then $f'(a)=0$.

\item A continuous function $f\colon\bbR^n\to\bbR^n$ satisfies $|(x-y)-(f(x)-f(y))|\leq\frac13|x-y|$ for every $x,y\in\bbR^n$. Prove that $f$ is surjective (onto).

\item A function $f\colon\bbR^n\to\bbR^n$ has the property that for every $k>0$ we have $|(x-y)-(f(x)-f(y))|\leq\frac{1}{k}|x-y|$ for every $x,y\in\bbR^n$ whose norm is less than $e^{-k}$. Prove that $f$ is differentiable at $0$ and compute its differential $f'(0)$.

\item A differentiable function $f\colon\bbR^n\to\bbR^k$ is said to be ``submersive'' at $0$ if $\rank f'(0)=k$. Assume such a function $f$ is submersive at $0$ and assume also that $f(0)=0$, and show that there is a function $g\colon\bbR^n\to\bbR^n$ which is defined, differentiable, and invertible near $0$, and so that $f(g(x_1,\ldots,x_n))=(x_1,\ldots,x_k)$. (In other words, every submersive function looks like the standard projection $\bbR^n\to\bbR^k$ near $0$).

\item \begin{enumerate}
  \item A subset $A\subset\bbR$ is known to have content $0$. Is it necessarily true that $\partial A$ also has content $0$?
  \item A subset $B\subset\bbR$ is known to have measure $0$. Is it necessarily true that $\partial B$ also has measure $0$?
  \end{enumerate}

\item If $f$ is a bounded function defined on a rectangle $R\subset\bbR^n$ and if $\supp f$ (the closure of $\{x\in R\colon f(x)\neq 0\}$) is a set of measure $0$, show that $f$ is integrable on $R$ and that $\int_Rf=0$.

\item If $f$ is a bounded non-negative function defined on a rectangle $R\subset\bbR^n$ and if $\int_Rf=0$, show
  \begin{enumerate}
  \item For every $b>0$, the set $\{x\in R\colon f(x)\geq b\}$ has content $0$.
  \item The set $\{x\in R\colon f(x)>0\}$ has measure $0$.
  \end{enumerate}

\item Let $f\colon\bbR\to\bbR$ be a smooth function. Prove that there is a smooth function $h\colon\bbR^2\to[0,1]$ such that $h(x,f(x))=1$ for every $x\in\bbR$, yet always, if $x,y\in\bbR$ and $|y-f(x)|\geq 1$, then $f(x,y)=0$.

\item Let $m\colon[0,1]\to M_{n\times n}(\bbR)$ be a path in the space of $n\times n$ matrices, and suppose that for every $t\in[0,1]$ the columns of $m(t)$ make a basis of $\bbR^n$. Show that the bases $m(0)$ and $m(1)$ define the same orientation of $\bbR^n$.

\item Show that if $F=\sum_i f_i(p)(p,e_i)$ and $G=\sum_i G_i(p)(p,e_i)$ are smooth vector fields on $\bbR^n$, then there is a third smooth vector field $H=\sum_i h_i(p)(p,e_i)$ on $\bbR^n$ such that
    \[ D_F\circ D_G-D_G\circ D_F = D_H, \]
    where $D_F\colon\Omega^0(\bbR^n)\to\Omega^0(\bbR^n)$ is the operation of directional derivative in the direction of $F$, which maps smooth functions on $\bbR^n$ to smooth functions on $\bbR^n$ (and likewise for $D_G$ and $D_H$).

\item An exploration problem: a 3-vector on $\bbR^4_{txyz}$ is a function $F\colon\bbR^4_{txyx}\to\bbR^3_{xyz}$. It can be regarded as a time-dependent vector field on $\bbR^3$, and so it makes sense to write $\grad$, $\curl$, and $\vdiv$ in this context, and also $\partial_t=\frac{\partial}{\partial t}$. Of course, you also need to consider ``scalar functions'' $f\colon\bbR^4\to\bbR$, to talk about $\grad$ and $\vdiv$. Can you interpret the sequence
    \[
      \Omega^0(\bbR^4) \overset{d}{\longrightarrow}
      \Omega^1(\bbR^4) \overset{d}{\longrightarrow}
      \Omega^2(\bbR^4) \overset{d}{\longrightarrow}
      \Omega^3(\bbR^4) \overset{d}{\longrightarrow}
      \Omega^4(\bbR^4)
    \]
    in this language of scalar functions, 3-vectors, $\grad$, $\curl$, $\vdiv$, and $\partial_t$?

\item Prove that the form $xdydz+ydzdx+zdxdy$ is closed but not exact on the 2-dimensional unit sphere $S^2\subset\bbR^3_{xyz}$.

\item $\omega$ is a smooth 3-form on $\bbR^7$, and we know that the integral of $\omega$ over every 3-cube in $\bbR^7$ vanishes. Prove that $\omega$ itself vanishes.

\item We will say that a 1-form $\omega$ on $\bbR^n$ is ``precise'' if its integral over any 1-cube depends only on the boundary of that 1-cube (namely, $\partial c_1=\partial c_2\Longrightarrow\int_{c_1}\omega=\int_{c_2}\omega$. Show that a 1-form $\omega$ is precise if and only if it is exact.

\item Suppose $M$ is a $k$-dimensional manifold in $\bbR^n$, and suppose $F$ is a smooth vector field on $M$ (so in particular $F(x)\in T_xM$ for every $x\in M$). Show that there is some vector field $G$ on some open set $A\supset M$ (in particular, $G(x)\in T_x\bbR^n$ for every $x\in A$) such that $G$ restricted to $M$ is $F$. You may need to use one of the precise definitions of a manifold, and something to make the local go global.

\item A smooth vector field $E$ defined on $\bbR^3$ is known to satisfy $\vdiv E=0$ outside of $D^3_{1/2}$, the 3-dimensional closed ball of radius $1/2$ in $\bbR^3$, and it is also known that $\int_{\partial D^3_1}(E\cdot n)dA=257$, where everything is taken with ``standard conventions'': orientations, positive normals, and area forms. Compute $\int_{\partial D^3_2}(E\cdot n)dA$ and $\int_{\partial D^3_1(p)}(E\cdot n)dA$, where $D^3_1(p)$ denotes the closed ball of radius $1$ about a point $p$, and $p$ is a point of $\partial D^3_2$.

\item A subset $B$ of $\bbR^3$ is the union two infinite lines positioned as on the figure on the left below, and in addition, oriented loops $R_1$, $R_2$, and $G_i$ for $i=1,2,3,4,5$ are also given as in the same figure. A vector field $F$ is also given, and it is known to be smooth away from $B$ and to satisfy $\curl F=0$ on $\bbR^3\setminus B$. It is known that $\int_{R_1}(F\cdot T)ds=\pi$ and $\int_{R_2}(F\cdot T)ds=e$. Compute $\int_{G_i}(F\cdot T)ds$ for $i=1,2,3,4,5$. {\em Hint.} You may want to also think about 2D subsets of $\bbR^3$ that are shaped like masks and/or tubes as in the figure below on the right
    \[
      \begin{array}{c} \includegraphics[width=3in]{../2021-257-AnalysisII/AConfiguration.png} \end{array}
      \qquad\qquad
      \begin{array}{c} \includegraphics[width=2in]{../2021-257-AnalysisII/MaskAndTube.png} \end{array}
    \]

\end{enumerate}

\noindent{\bf Please watch this page for changes --- I may add to it later.}\hfill{\footnotesize Last modified: \today, \currenttime}
\vfill

\end{document}
