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\title{Analysis problem set -- Integration workshop 2016}
%\author{}
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\begin{document}
\maketitle
%\section{}
%\subsection{}

\newcounter{excount}



\section{Sequences and series}

\begin{enumerate}




%\item
%Given a set $S$ in a metric space, let $\bar{S}$ denote 
%the set obtained by adding all the limit points of $S$ to S, 
%i.e., $p \in \bar{S}$ if there is a sequence
%$x_n \in S$ such that $x_n \ra p$. 
%Show that $\bar{\bar{S}} = \bar{S}$. 
%\medskip


%(b) Show that the set of points of discontinuity of an arbitrary real
%valued function is always a $F_\sigma$ set and the set of points of continuity is
%always a $G_\delta$ set.
%
%(c) [Optional.. hard..] There is no real valued function that is continuous
%on all the rationals but discontinuous on all the irrationals (Use
%Baire category theorem).
%\medskip

\item $a_n$ is a non-negative, decreasing sequence, i.e. $a_n \geq a_{n+1} \geq 0$ for all $n$. Show that $\sum_n a_n$ converges if and only if $\sum_k 2^k a_{2^k}$ converges. What can you conclude about $\sum_n \frac{1}{n \log \log n}$.

\item
Assume that $a_n > 0$ and $b_n > 0$ for n = 1, 2, . . ., and suppose that 
$\lim_{n \ra \infty} a_n/b_n =1$. Prove that 
$\sum_{n=1}^\infty a_n$ converges if and only if $\sum_{n=1}^\infty b_n$
converges.

\item Show that the product $\displaystyle{\prod_n (1+a_n)}$ converges (resp. converges absolutely) if and only if the series $\displaystyle{\sum a_n}$ converges (resp. converges absolutely).

\item Let $x_n$ be a bounded sequence of real numbers. 
Let $x=\limsup_{n \ra \infty} x_n$. 
\begin{enumerate}
\item Prove that there is a subsequence $x_{n_k}$ such that 
$\lim_{k \ra \infty} x_{n_k} = x$. 
\item Prove that for every subsequence $x_{m_k}$ of $x_n$ which
converges, $\lim_{k \ra \infty} x_{m_k} \le x$ 
\end{enumerate}

\item A non-negative sequence $a_n$ is {\em subadditive} if $a_{n+m} \leq a_n + a_m$ for all $m,n \in \mathbb{Z}^+$. The goal of this problem is to show that, for any subadditive sequence, the limit $\displaystyle{\lim_{n \to \infty} \frac{a_n}{n} }$ exists.

(a) Show that, for any $n \in \mathbb{Z}^+$, there is a constant $C_n$ such that, for all $k \geq n+1$, we have 
$$
\frac{a_k}{k} \leq \frac{a_n}{n} + \frac{C}{k}.
$$

(b) Show that $\displaystyle{ \limsup_{k \to \infty} \frac{a_k}{k} \leq  \liminf_{n \to \infty} \frac{a_n}{n}}$. Why is this enough to prove the claim?
 
 
\item $A_n$ is a sequence of subsets of $\reals$. 

(a) Explain why the following definitions make sense:
$$
\limsup_n A_n = \bigcap_{j= 1}^\infty \bigcup_{n= j}^\infty A_n = \{x | x \in A_n \mbox{ \em infinitely often}\}
$$
and 
$$
\liminf_n A_n = \bigcup_{j= 1}^\infty \bigcap_{n= j}^\infty A_n = \{x | x \in A_n \mbox{ \em eventually}\}
$$

(b) What can you conclude about a sequence of subsets for which $\limsup_n A_n = \liminf_n A_n$. Explain why this is a very restrictive definition of a limit for sets. In your classes you will explore alternative (and less restrictive) notions for limits of sets.

\item
(a) Let $f$ be a continuous function on $[0,1]$. 
Show that the following limits exist and evaluate them:
\beann
\lim_{n \ra \infty} \int_0^1 \, x^n \, f(x) \, dx
\eeann
\beann
\lim_{n \ra \infty} n \int_0^1 \, x^n \, f(x) \, dx
\eeann

(b) Let $g$ be a differentiable function on $[0,1]$ such that $g(1)=0$. 
Show that the following limit exists and evaluate:
\beann
\lim_{n \ra \infty} n^2 \int_0^1 \, x^n \, g(x) \, dx
\eeann
\medskip

\item Consider the alternating series
$$
1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots
$$
(a) Show that, as presented the series (i.e the sequence of partial sums taken with the given ordering) converges to $\ln(2)$.\\
(b) Use a computer to estimate the result from rearrangement
$$
1 + \frac{1}{3} - \frac{1}{2} + \frac{1}{5} +  \frac{1}{7} - \frac{1}{4} + \cdots
$$
Is this also $\ln(2)$. If not, can you guess what the answer is? \\
(c) Prove the rearranged series indeed converges to the answer that you claimed in the previous part. \\
(d) Show that, given any pair of real numbers $a < b$, the terms in the series can be rearranged so that the sequence $x_n$ of partial sums (of the rearranged series) satisfies 
$$
\limsup_n x_n \geq b, \qquad \liminf_n x_n \leq a.
$$

%\item Let 
%$$
%f(x) = \begin{cases} a & x = 0 \\ x^{-x} & x > 0 \end{cases}
%$$
%(a) Show that there is an appropriate choice for $a$ so that $f$ is continuous on $[0,1]$.
%
%(b) For this choice of $a$, show, with justification, that 
%$$
%\int_0^1 f(x) dx = \sum_{n = 1}^\infty n^{-n}.
%$$
%Possibly useful integral: If $\beta > 0$ and $n \in \mathbb{N}$, 
%$$
%\int_0^\infty x^n e^{-\beta x} dx = \frac{n!}{\beta^{(n+1)}} 
%$$  
\item
Give a counter example to the following statement: A sequence of
differentiable functions $f_n$ converges uniformly to a 
differentiable function $f$. 
This implies that the sequence of derivatives $f_n^\prime$
converges pointwise to $f^\prime$.
\medskip

\item
$f_n$ is a sequence of uniformly bounded non-negative Riemann integrable
functions. For each $x \in [0,1]$, the real sequence $f_k(x)$ is monotone non-decreasing, i.e 
$n \ge m$ and $x \in [0,1]$ implies that $f_n(x) \ge f_m(x)$, 
and $0 \le f_n(x) \le  K < \infty$ for all $n, x$.

(a) Show that the sequence $f_n(x)$ converges pointwise to a bounded
function $f(x)$.

(b) Show that the sequence of real numbers 
$s_k = \int_0^1 \, f_k(x)\, dx$ also converges to some number $s$.

(c) Is it true that $f$ is Riemann integrable, and
$\int_0^1 f(x) \, dx = s$ ?
Prove or give a counter example.
\medskip


\end{enumerate}

\section{Inequalities}

\begin{enumerate}

\item Prove the inequality $e^x \geq 1+x$ for all $x \in \reals$.

\item A $C^2$ function $f : \mathbb{R} \to \mathbb{R}$ is convex if $f''(x) > 0$ for all $x$. Show that, for any finite collection of points $a_1,a_2,\ldots,a_n$, we have the inequality
$$
\frac{f(a_1)+f(a_2) + \cdots + f(a_n)}{n} \geq f\left(\frac{ a_1+a_2+ \cdots+a_n}{n}\right).
$$
(Hint: Draw a picture. Start with $n = 2$)

\item Let $p,q$ be such that $\displaystyle{\frac{1}{p} + \frac{1}{q} = 1}$, and $a,b > 0$ be real numbers. Prove Young's inequality
$$
\frac{a^p}{p} + \frac{b^q}{q} \geq a b
$$ 
(Hint: The function $f(x) = - \log(x)$ is convex.)


\end{enumerate}

\section{Complex analysis}

\begin{enumerate}

\item
Let $f(z)$ be analytic at $c$. Write $f^\prime(c)=r e^{i \theta}$. 
Write $z=x+iy$ and 
$f(z)=u(x,y)+iv(x,y)$. We can think of $f$ as a map $(u(x,y),v(x,y)$ 
from $\reals^2 \ra \reals^2$. The total derivative of this map at $c$ 
is a two by two matrix. Find it in terms of $r$ and $\theta$. 
Express the direction derivatives of the map on $\reals^2$ in terms 
of $r$ and $\theta$. 
\medskip

\item
Define
\beann
f(z) = \sum_{k=1}^\infty a_k (z-z_0)^k
\eeann

\noindent (a) If the radius of convergence of the above power series is $r$, 
then show that the radii of convergence for the series obtained by 
differentiating and integrating the above series termwise is also $r$.

\noindent (b) Show that the termwise differentiated series converges 
uniformly on every disk of the form $|z - z_0 | \le \rho < r$. 
Use this to show that $f(z)$ is given by termwise differentiating the 
above series, for any compact subset of the disk $|z - z_0 | < r$.



\item
Suppose that $\gamma$ is a piecewise smooth positively, 
counterclockwise oriented, simple closed curve. 
Use Green's Theorem to show that the value of the integral
\beann
\oint_\gamma {dz \over z-p} 
\eeann
equals $0$ if $p$ is outside $\gamma$ and $2\pi i$ if $p$ is inside 
$\gamma$.






\item Let $f(z)$ be analytic at $z_0$. Prove that for sufficiently small
$\epsilon$, 
\beann
{1 \over 2 \pi i} \oint_{|z-z_0|=\epsilon} {f(z) \over (z-z_0)^n} \, dz 
= {f^{(n-1)}(c) \over (n-1)!}
\eeann
The contour is the circle centered at $z_0$ with radius $\epsilon$ 
traversed in the counterclockwise direction.
This is a standard theorem in complex analysis books. 
The exercise is to prove it directly from the statement of 
Cauchy's theorem given in the notes. Hint: power series. 

\item 
If $f(z)$ is analytic on the closed disk $B(z_0 ,r)$, 
show that there is a constant $C$ such that 
the derivatives of $f$ at $z_0$ can be bounded by
\beann
\left|f^{(n)}(z_0)\right| \le {C n! \over r^{n+1} }
\eeann
\medskip

\item 
For each of the following functions $f$ and $z_0$, determine 
if the function has a pole or essential singularity at $z_0$. 
In the case of a pole determine the order of the pole and 
the principal part. 

\smallskip
\noindent (a) $f(z)={1 \over z \sin(z)}$, $z_0=0$

\smallskip
\noindent (b) $f(z)={1 \over (z^2+1)^2}$, $z_0=i$

\smallskip
\noindent (c) $f(z)=\exp(-1/z)$, $z_0=0$. 

\smallskip
\noindent (d) $f(z)=\tan^{2}(z)$, $z_0=\pi/2$. 
\medskip

\item
A M\"obius transformation, or a fractional linear transformation, is a
mapping of the form
\beann
w = f (z) = {az + b \over cz + d}
\eeann    
where $a, b, c, d \in \complex, ad -bc \neq 0$. 
This can be extended to the Riemann sphere = $\complex \cup \{\infty\}$
\beann
f (\infty) = a/c,  \quad   f (-d/c) = \infty
\eeann

\noindent (a) Show that the composition of M\"obius transformations is also a M\"obius transformation..

\noindent   (b) Show that the M\"obius transformations map circles to circles on
the Riemann sphere (Note: A straight line on $\complex$ is considered a circle
through $\infty$ on the Riemann sphere).
\medskip


\end{enumerate}


\section{Metric spaces and continuous functions}
\begin{enumerate}

\item Show that $d(x,y) = \log(1+|x-y|)$ defines a metric on $\mathbb{R}$. (Hint: Use ideas from the section on inequalities!)


%\item \label{prob:boundedmetric} For $x,y \in \reals$ let 
%\beann
%d(x,y) = \frac{|x-y|  }{ 1 + |x-y|}
%\eeann
%\begin{enumerate}
%\item Show this is a metric.
%\item Does this metric give $\reals$ a different topology from the one that
%comes from the usual metric on $\reals$? You should prove your answer. 
%\end{enumerate}
%  
%\item {\bf The structure of open sets in $\mathbb{R}$.}
%
%In what follows, we are considering the standard (metric) topology on 
%$\mathbb{R}$.
%
%\begin{enumerate}
%
%\item Let $S$ be a nonempty open subset of $\R$. For each $x \in S$, 
%let $A_x = \{a \in \R : (a,x] \subseteq S\}$ and 
%$B_x = \{b \in \R : [x,b) \subseteq S\}$. Show that, $A_x$ and 
%$B_x$ are both non-empty.
%
%\item Where $x \in S$ as above, if $A_x$ is bounded below, let 
%$a_x = \inf(A_x)$. Otherwise, let $a_x = - \infty$, and define 
%$b_x$ is a corresponding manner. Show that $x \in I_x = (a_x,b_x) \subseteq S$.
%
%\item Show that $S = \cup_x I_x$.
%
%\item Show that the intervals $I_x$ give a partition of $S$, {\em i.e.}, 
%for $x, y \in S$, either $I_x = I_y$ or $I_x \cap I_y = \emptyset$.
%
%\item Show that the set of distinct intervals $\{I_x : x \in S\}$ is countable. 
%
%\item Prove that every open set in $\R$ is a countable disjoint union of 
%open intervals.
%
%\end{enumerate}
%
% 
%\item If $f:(X,\mathcal{T}) \rightarrow (Y,\mathcal{S})$ and
%  $g:(Y,\mathcal{S}) \rightarrow (Z,\mathcal{V})$ are continuous, show
%  that the composition $g \circ f :(X,\mathcal{T}) \rightarrow
%  (Z,\mathcal{V})$ is also continuous.
%  
%\item A subset $A$ of $X$ is called dense if the closure of $A$ is $X$. 
%A topological space $X$ is called {\em separable} 
%if there exists a countable dense subset. 
%
%\begin{enumerate}
%\item Prove that $\reals^n$ with the usual topology is separable. 
%
%
%
%\item Since the rationals are a dense set in $\reals$ does it follow that every open subset of $\reals$ is determined by the rational elements of the set?
%
%\item Show that the collection of all intervals of the form $(r_1,r_2)$ where both $r_1$ and $r_2$ are rational is countable, and further, every open subset of $\reals$ is uniquely determined by the intervals with rational endpoints that are contained in it.
%
%\item Is the collection of all open subsets of $\reals$ countable?
%
%\item %Recall the definition of $l^p(\naturals)$ with $1 \le p \le \infty$.  For what $p$ is this metric space separable? Prove your answer. 
%If a metric space is separable, show that there is a countable collection of open balls (i.e sets of the form $B_\epsilon(x) = \{y: d(x,y) < \epsilon\}$) such that every open set can be written as a union of balls from this collection.
%  
%\end{enumerate}
%
%\item {\bf Accumulation points}  $A \subseteq \R$, and $A'$ denotes the 
%set of all the accumulation points of $A$.
%\begin{enumerate}
%\item If $y \in A'$ and $U \subseteq \R$ is an open set containing $y$, 
%show that there are infinitely many distinct points in $A \cap U$.
%\item Show that 
%$$
%A' = \bigcap_{x \in A} \cl(A\backslash\{x\}).
%$$
%\item Using this, or otherwise, show that $A'$ is a closed set.
%\item Show that $\cl(A) = A \cup A'$.
%\end{enumerate}

%%\begin{enumerate}
%%\item 
%$(X,T)$ is a topological space and the topology $T$ is Hausdorff. $A \subseteq X$. Let $A'$ denote the set of accumulation points of $A$. Show that 
%$$
%A' =  \bigcap_{x \in A} \overline{A \backslash\{x\}}.
%$$
%\end{enumerate}

%\item {\bf Boundary points}
%%\begin{enumerate}
%\item Problem 3.5.30, Pg. I-150 of Prof. Flaschka's notes.
%\item Show that the above definition of boundary point agrees with the definition in class, {\em viz.}, $x \in X$ is a boundary point of $A$ iff for every neighborhood $U$ of $x$, $U \cap A \neq \emptyset$ and $U \cap A^c \neq \emptyset$.
%\end{enumerate}



\item 
\begin{enumerate}
\item
Show that $\R$ and $(0,1)$ are homeomorphic, i.e., there is  continuous
bijection between them whose inverse is also continuous.

\item
Let $f : (0,1) \rightarrow \R$ be your homeomorphism. Show there 
is a Cauchy sequence $x_n$ in $(0,1)$ such that $f(x_n)$ is not 
Cauchy in $\R$. 
\end{enumerate}

%\item In the lectures we stated a proposition that said that if 
%$(X,d)$ and $(Y,d^\prime)$ are metric spaces and $f:X \rightarrow Y$,
%then the $\epsilon-\delta$ definition of continuity of $f$ and the open set 
%defintion are equivalent. Prove this proposition.
%
%\item Let $C((0,1))$ be the set of bounded continuous functions on 
%$(0,1)$ with the usual sup norm. So 
%\beann
%d(f,g) = \sup_{0 < x < 1} |f(x)-g(x)|
%\eeann
%Let $U=\{f: f(x) > 0 \quad \forall x \in (0,1)\}$, 
%and $V=\{f: f(x) \ge 0 \quad \forall x \in (0,1)\}$. 
%For each of $U$ and $V$ determine if the set is open or closed or 
%neither. You should prove your answer. 

\item
(a) Define a function $f$ on $[0,1]$ by
$$
f(x) = \begin{cases} 0 & x \mbox{ is irrational} \\ 1 & x \mbox { is rational} \end{cases}
$$
Identify the points where $f$ is continuous and the points where $f$ is discontinuous.

(b) Define a function $g$ on $[0,1]$ by
$$
g(x) = \begin{cases} 0 & x \mbox{ is irrational} \\ \frac{1}{q} & x = \frac{p}{q} \mbox{ expressed in its lowest terms} \end{cases}
$$
Identify the points where $f$ is continuous and the points where $f$ is discontinuous.

(c) (Optional.. hard) Is there an example of a function $f : [0; 1] \ra \reals$ that is 
discontinuous at all the irrationals and continuous at all the rationals?

\item A function $f:\reals \to \reals$ is {\em uniformly continuous} if for all $\epsilon > 0$, there is a $\delta > 0$ such that {\em for all } $ x \in \reals$, $|y-x| < \delta \implies |f(y) - f(x)| < \epsilon$. This means we can pick a $\delta$ ``that works" for all points in $\reals$. Clearly, this definition also extends to general metric spaces.
 
(a) Let f be a continuous real valued function on $[0,\infty)$ such that 
$\lim_{x \ra \infty} f(x)$ exists (and is finite). Prove that $f$
is uniformly continuous on $[0,\infty)$.

(b) Let $f: \reals \to \reals $ be uniformly continuous. If $x_n$ is a Cauchy sequence, show that $f(x_n)$ is also a Cauchy sequence. Is the converse true?

\item Let $(M,d)$ be a metric space. Define a set $\tilde{M} \subset M^\mathbb{N}$ as the collection of all the Cauchy sequences in $(M,d)$.

(a) Show that $\displaystyle{\rho(\{x_n\},\{y_n\}) = \lim_{n \to \infty} d(x_n,y_n)}$ is a well defined function on $\tilde{M} \times \tilde{M}$, i.e the limit always exists.

(b) Show that $\rho$ satisfies the triangle inequality.

(c) Let $a_k = \{x_n\}_k$ be a sequence in $\tilde{M}$ (i.e a sequence of Cauchy sequences in $(M,d)$). Further, assume that $a_k$ is a Cauchy sequence with respect to $\rho$, i.e. for all $\epsilon > 0$, there is an index $K$ such that for all $j,k > K$, we have 
$$
\rho(a_j,a_k) = \lim_{n \to \infty} d((x_n)_j, (x_n)_k) < \epsilon.
$$
Construct an element $a^*$ in $\tilde{M}$ such that $\rho(a_k,a^*) \to 0$ as $k \to \infty$.

\medskip


\end{enumerate}


\section{Compactness}

\begin{enumerate}

%\item Let $C([0,1])$ be the set of bounded continuous functions on 
%$[0,1]$ with the usual sup norm. So 
%\beann
%d(f,g) = \sup_{0 \le x \le 1} |f(x)-g(x)|
%\eeann
%Given a function $g \in C([0,1])$, 
%let $U_g=\{f: f(x) > g(x) \quad \forall x \in [0,1]\}$. 
%Prove that $U$ is open.  

\item Prove or disprove the following:
\begin{enumerate}
\item $A$ is finite and $U$ is a open subset of $\R$. 
If $A \subseteq U$, there exists an $\epsilon > 0$ such that for 
all $x \in A$, $N(x,\epsilon) \subseteq U$.
\item $P$ is countable and $U$ is a open subset of $\R$. If $P \subseteq U$, 
there exists an $\epsilon > 0$ such that for all $x \in P$, 
$N(x,\epsilon) \subseteq U$.
\item $F$ is closed and $U$ is a open subset of $\R$. 
If $F \subseteq U$, there exists an $\epsilon > 0$ such that for all 
$x \in F$, $N(x,\epsilon) \subseteq U$.
\item $K$ is compact and $U$ is a open subset of $\R$. If $K \subseteq U$, 
there exists an $\epsilon > 0$ such that for all $x \in K$, 
$N(x,\epsilon) \subseteq U$.
\end{enumerate}

%\item This is a continuation of problem~2.\ref{prob:boundedmetric}. For $x,y \in \reals$, let 
%\beann
%d(x,y) = \frac{|x-y|  }{ 1 + |x-y|}
%\eeann
%We already showed that $d$ is a metric on $\reals$.
%\begin{enumerate}
%\item Prove that with this metric, the entire space of $\reals$ is not compact
%even though it is closed and bounded. 
%\item Can you characterize all the compact sets in $(\reals,d)$ ?
%\end{enumerate}
  
%\item In a metric space $(X, d)$, a sequence a1, a2, ... of points of 
%X is called a Cauchy sequence
%if for each " > 0 there is a positive integer N such that 
% d(an, am) < " whenever n,m > N.
%A metric space is called complete if every Cauchy sequence in X 
%converges to a point of X. Show that a compact metric space is complete.

\item Let $f:(0,\infty) \to \mathbb{R}$ be a continuous function such that the following limits exist:
$$
\lim_{x \to 0} \frac{f(x)}{x} = \alpha > 0, \quad \lim_{x \to \infty} \frac{f(x)}{x^2} = \beta < 0.
$$
Show that, there is a finite constant $C$ such that $|f(x)| \leq C(x + x^2)$ for all $x \in (0,\infty)$.


\item A metric space $X$ is said to be {\em totally bounded} if for 
any $\epsilon > 0$ it can be covered by finitely many $\epsilon$-balls. 
Also, a metric space is complete, if every Cauchy sequence in $X$ converges. 
Prove that $X$ is compact if and only if $X$ is complete and totally bounded.

\item Define the ``distance'' between two subsets of $\reals$ by $\displaystyle{\rho(A,B) = \inf_{x \in A, y \in B} |x-y|}$.

(a) Is $\rho$ a metric on the power set $2^\reals$ (i.e the collection of all subsets of $\reals$)?

(b) If $A$ is closed and $B$ is compact, show that $\rho(A,B) = 0$ if and only if $A \cap B$ is nonempty.

(c) If $A$ and $B$ are closed, does it follow that $\rho(A,B) = 0 \implies A \cap B \neq \emptyset$?

%\item $l^2$ is the space of square summable sequences of real 
%numbers, i.e., $(x_n)_{n=1}^\infty$
%such that $\sum_{n=1}^\infty x_n^2 < \infty$. The norm is defined by 
%\begin{equation}
%|| (x_n)_{n=1}^\infty|| = \left[ \sum_{n=1}^\infty x_n^2  \right]^{1/2}
%\end{equation}
%\begin{enumerate}
%\item Prove the interior of every compact subset of $l^2$ is empty. 
%\item Define
%\begin{equation}
%F= \{ (x_n)_{n=1}^\infty : \sum_{n=1}^\infty \, n \, x_n^2 \le 1\}
%\end{equation}
%Prove that $F$ is sequentially compact. 
%\end{enumerate}

\end{enumerate}

%\section{Calculus}
%
%\begin{enumerate}
%
%\item Prove the inequality $e^x \geq 1+x$ for all $x \in \reals$.
%
%\item $f: \reals^k \to \reals^m$ and $g :\reals^m \to \reals^n$ are differentiable. Prove the chain rule for the composition $g \circ f$. 
%
%\item A $C^2$ function $f : \mathbb{R} \to \mathbb{R}$ is convex if $f''(x) > 0$ for all $x$. Show that, for any finite collection of points $a_1,a_2,\ldots,a_n$, we have the inequality
%$$
%\frac{f(a_1)+f(a_2) + \cdots + f(a_n)}{n} \geq f\left(\frac{ a_1+a_2+ \cdots+a_n}{n}\right).
%$$
%(Hint: Draw a picture. Start with $n = 2$)
%
%\item
%(a) Give an example of a function of two variables that is discontinuous
%at the origin, but whose partial derivatives at the origin exist.
%
%(b) Give an example of a function of two variables all of whose directional
%derivatives exist at the origin, but the function itself is not
%differentiable at the origin.
%\medskip
%
%\item An open subset $O \subset \reals^n$ has the property that for any pair of points $x,y \in O$, there is a differentiable function $\gamma:[0,1] \to O$ such that $\gamma(0) = x$ and $gamma(1) = y$. 
%$f : O \ra \reals^m$ is differentiable and $Df = 0$ on $O$. Show that $f$ is a constant on $O$.
%
%(The Hypothesis can be slightly weakened. It is sufficient to assume that $O$ is connected and open.)
%\medskip
%
%\item
%Evaluate the derivatives of the following matrix functions:
%
%(a) $inv : GL(n) \ra GL(n)$ given by $inv(M) = M^{-1}$.
%
%(b) The determinant function which maps $GL(n)$ to $\reals$. 
%\medskip
%
%
%\item
%Show that every point $p$ on the sphere $x^2 + y^2 + z^2 = 1$ 
%has a (3 dimensional) neighborhood $U$ such that there is a smooth, one to one
%mapping of an open neighborhood $V$ of the origin in $\reals^3$ such that the
%plane $z = 0$ maps to the surface of the sphere.
%\medskip
%
%\item
%Let $f$ and $\frac{\partial f }{ \partial y}$ be continuous on 
%$[0, 1] \times [0, 1]$ and assume that $p, q : [0, 1] \ra [0, 1]$ 
%are differentiable.
%Define
%\beann
%F(y) = \int_{p(y)}^{q(y)} \, f(x, y) dx, \quad  y \in  [0, 1]
%\eeann
%
%Use the chain rule to find $F^\prime(y)$. Hint: Consider 
%$G(x_1, x_2, x_3) = \int_{x_1}^{x_2} \, f(t, x_3) dt$.
%\medskip
%
%\item
%Let $f(x, y) = (x^2 - y^2)/(x^2 + y^2)^2$. Show that
%\beann
%\int_0^1 \left( \int_0^1 f(x,y) dx \right) dy = -\pi/4, \quad but \quad 
%\int_0^1 \left( \int_0^1 f(x,y) dy \right) dx = \pi/4
%\eeann
%\medskip
%
%\item
%Show that in a neighborhood of the origin, the systems of equations
%\beann
%3x + y - z + u^2 &=& 0 \\
%x - y + 2z + u &=& 0 \\
%2x + 2y - 3z + 2u &=& 0 \\
%\eeann
%can be solved for $x, y, u$ in terms of $z$; 
%for $x, z, u$ in terms of $y$; 
%for $y, z, u$ in terms of $x$.
%What can you say (using the implicit function theorem)
%about solving them  for
%$x, y, z$ in terms of $u$?
%\medskip
%
%\item
%Let $F$ denote the vector field $(x^2 - z^2, 2xy, z)$ on $\reals^3$. 
%Compute in two different ways the surface integral
%\beann
%\int \int_T \, F \cdot n  \, dA
%\eeann
%where $T$ denotes the surface of the tetrahedron bounded by 
%$x \ge 0, y \ge 0, z \ge 0, x + y + z  \le 1$ 
%and $n$ denotes the outward normal to $T$.
%Use the two answers to verify the divergence theorem.
%\medskip
%
%\item
%Compute in two different ways the line integral
%\beann
%\oint \, ydx - xdy + z^2 dz
%\eeann
%where $C$ is the intersection of the paraboloid $z = x^2 + 4y^2$ with the
%cylinder $x^2 + y^2 = 9$, traversed counter-clockwise when viewed from
%the point $(0,0,100)$. Use the two answers to verify Stokes' theorem
%\medskip
%
%\item 
%A smooth function $F$ on the (open) unit disc in $\reals^2$ satisfies $F(0,0) = (0,0)$ and $\|F(x)\| \geq 3 \|x\|$. 
%
%(a) Show that $F$ is one to one in a neighborhood of the origin. 
%
%(b) Let $F = (u,v)$ in components. Show that, for all $a < 1$, the integral
%$$
%\frac{1}{2\pi} \oint_{\|r\|=a} \frac{u \nabla v - v \nabla u}{u^2+v^2} \cdot dr
%$$
%is an integer and this value does not depend on $a$.
%
%(c) Can you determine the value of the integral in terms of the Jacobian $A = DF(0,0)$ of the mapping $F$ at the origin?
%
%(d) Can you construct a smooth function $F$ on the (open) unit disc in $\reals^2$ satisfying $F(0,0) = (0,0)$, $F(x) \neq (0,0)$ for $x \neq (0,0)$, such that the value of the integral is 2? 
%
%\item $M: \mathbb{R}^2 \to \mathbb{R}^2$ is the mapping 
%$$
%M(x,y) = (2 x - \sin(xy), \frac{1}{3}y + x^2-y^2).
%$$
%
%(a) Show that $M$ is invertible in a neighborhood of the origin.
%
%(b) The {\em unstable manifold} of the origin is the set $\Gamma_u = \{(x,y) | M^{-n}(x,y) \to (0,0) \mbox{ as } n \to \infty\}$. Show that $\Gamma_u$ is invariant under $M$.
%
%(c) Show that, there is a unique function $f$ such that $f(x) \to 0$ when $x \to 0$, and the unstable manifold is given by the graph of the function $f$ for sufficiently small $x$, i.e.
%$$
%\Gamma_u \cap B_{\epsilon}(0,0) = \{(x,f(x)) | |x| < \epsilon\}  \cap B_{\epsilon}(0,0)
%$$
%
%\end{enumerate}
%


\end{document}

\end


