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\begin{document}
\bibstyle{ams}

\newcounter{excount}

\centerline{\Large \bf Integration Workshop 2011} 

\bigskip

\centerline{\Large \bf Calculus/Analysis Problems} 

\section{Calculus}


\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
$f : O \ra \reals^m$ is differentiable and $Df = 0$ on $O$, an open subset of 
$\reals^n$. Is $f$ necessarily a constant on $O$. What condition do we need to have on $O$ so that $f$ is constant?
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Let $f$ and ${\partial f \over \partial y}$ 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

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Let $f : \reals^n \times \reals^m \ra \reals^p$. We say $f$ is bilinear if
for $x,x_1,x_2 \in \reals^n$, for $y,y_1,y_2 \in \reals^m$ and $a \in \reals$
we have 
\beann
f(ax,y) &=& a f(x,y) = f(x,ay) \\
f(x_1 + x_2,y) &=& f(x_1,y) + f(x_2,y) \\
f(x,y_1 + y_2) &=& f(x,y_1) + f(x,y_2) 
\eeann
Prove that if $f$ is bilinear then $f$ is differentiable and 
express its derivative in terms of $f$. 
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
(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

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Show that 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 $x$; but not for
$x, y, z$ in terms of $u$.
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Let $f(x, y) = (x - y)/(x + y)^2$. Show that
\beann
\int_0^1 \left( \int_0^1 f(x,y) dx \right) dy = -1/2, \quad but \quad 
\int_0^1 \left( \int_0^1 f(x,y) dy \right) dx = 1/2
\eeann
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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 \mathbf{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

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Compute in two different ways the line integral
\beann
\oint_C \, 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

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Let $U \subset \reals^3$ be open, $f$ a differentiable function 
on $U$ and $F$ a differentible vector field on $U$. 
Define the forms
\beann
\omega^1_F &=& F^1 dx + F^2 dy + F^3 dz \\
\omega^2_F &=& F^1 dy \wedge dz + F^2 dz \wedge dx + F^3 dx \wedge dy
\eeann

\noindent (a) 
Show that 
\beann
df &=& \omega^1_{\nabla f}, \\
d \, \omega^1_F &=& \omega^2_{\nabla \times F} \\
d \, \omega^2_F &=& (\nabla \cdot F) \, dx \wedge dy \wedge dz 
\eeann

\noindent (b) Use (a) to show how Green's theorem, the divergence 
theorem and the classical Stoke's theorem are all special cases of
the general Stoke's theorem. 

\noindent (c) Use (a) to show that 
\beann
\nabla \times (\nabla f) &=& 0 \\
\nabla \cdot (\nabla \times F) &=& 0 \\
\eeann

\noindent (d) Use (c) to show that if $F$ is a vector field on a 
convex open set $U$ and $\nabla \times F=0$, then there is a 
differentiable function $f$ on $U$ such that $F = \nabla f$.
Similarly, if $\nabla \cdot F=0$ on $U$, show that there is a vector 
field $G$ on $U$ with $F= \nabla \times G$. 
\medskip

\section{Real Analysis} 

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Assume that $a_n > 0$ and $b_n > 0$ for n = 1, 2, . . ., and suppose that 
$\lim_{n \ra \infty} a_n/b_n =1$. Then
$\sum_{n=1}^\infty a_n$ converges if and only if $\sum_{n=1}^\infty b_n$
converges.
\medskip


\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Test for convergence ($p, q$ and $r$ are fixed real numbers.)
$$\sum_{n=1}^\infty n^k e^n \qquad\qquad \sum_{n=1}^\infty (\log n)^p 
\qquad\qquad \sum_{n=1}^\infty p^nn^p\qquad\qquad
\sum_{n=1}^\infty \frac{1}{n^p - n^q}\qquad\qquad \sum_{n=1}^\infty \frac{1}
{p^n - q^n},$$
$$\sum_{n=1}^\infty \frac{p^n}{n^q(\log n)^r},\qquad\qquad
 \sum_{n=1}^\infty \frac{(2n)!(3n)!}{n!(4n)!}, \qquad\qquad
\sum_{n=1}^\infty \frac{1}{ n^{(1 + \frac{1}{n})}}.$$
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Let $a_n$ be a bounded real sequence. Let $A$ be the set of all real numbers $a$
such that there is a subsequence of $a_n$ which converges to $a$. 
Prove that 
\beann
\sup A = \limsup_{n \ra \infty} a_n
\eeann
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Let $f_n$ be uniformly continuous on $S$, $f$ continuous on $S$ and suppose
$f_n$ converges uniformly to $f$. Does it follow that $f$ is uniformly
continuous? Either prove that it does or give a counter example. 
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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 $[0,\infty)$.
\medskip


\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Show that $\rationals[x]$, the set of all polynomials in 
one variable with rational coefficients is countable.
\medskip


\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
(a) Give an example of a function $f : [0; 1] \ra \reals$ that is 
continuous at all the irrationals and discontinuous at all the rationals.

(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


\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
(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


\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Give a counterexample 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

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
$f_n$ is a sequence of uniformly bounded non-negative Riemann integrable
functions which are monotone non-decreasing on $[0,1]$, 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 $n$ and $x$.

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

(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 counterexample.
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Let $\cal F$ be the set of continuous functions on $S$ which have a 
continuous derivative on $S$. Define a metric $d$ on this set by 
\beann
d(f,g) = ||f-g||_\infty + ||f^\prime-g^\prime||_\infty
\eeann
Is this a complete metric space? 

\medskip

\section{Complex Analysis} 

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Let $f$ be analytic at $c$. Write $f^\prime(c)=r e^{i \theta}$, $z=x+iy$ 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 $2\times 2$ matrix. Find it in terms of $r$ and $\theta$. 
Express the directional derivatives of the map on $\reals^2$ in terms 
of $r$ and $\theta$. 
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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 $|z - z_0 | < r$.
\medskip


\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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 
$\gamma$.
\medskip


\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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. 
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
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 $a, b, c, d \in \complex$. 
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 Mobius transformations form a group.

\noindent   (b) Show that the Mobius 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

\section{Differential Equations}

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Derive a series representation for the fundamental matrix $\Phi(t)$ of the
linear system $y = Ay$ using Picard iteration.
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Solve $y^\prime = Ay + g(t)$ with $y(0) = (0, 0)$, and 
\beann
A=\pmatrix{0 & 1 \cr -1 & 0}, \quad g(t)=\pmatrix{ 2 e^{-t} \cr t e^{-2t}} 
\eeann
\medskip

\addtocounter{excount}{1}
\noindent {\bf \arabic{excount}.}
Let $J_0$ be an $n \times n$ matrix that has the value $1$ for entries on
the superdiagonal and $0$ elsewhere.

   (a) Find $\exp(tJ_0 )$.

   (b) Verify that the identity matrix $I$ and $J_0$ commute.

   (c) Use this to find $\exp(t(\lambda I + J_0 ))$.

   (d) Describe the unique solution to the differential equation 
     $y^\prime = Ay$ where $A$ is a matrix in Jordan canonical form.
\medskip

\end{document}

\end