\documentclass[10pt]{math25}
% By Tom Braden
% Minor editing by Nils Barth

\title{Math 25b --- Problem Set 11, for review (not due)}
\begin{document}

Reminder: the final exam is Thursday, May 14 at 9:15am in Science Center E.
There will be a review session on Monday May 11th at the usual place and time.

\begin{enumerate}

\item Let $\gamma$ be a piecewise $\cC^1$ path which first travels the
quarter circle centered at the origin from $(1,0)$ to $(0,1)$ counterclockwise,
and then goes in a straight line from $(0,1)$ to $(-1,0)$.  Calculate the
integral $\int_\gamma 2xy\,dx - y^2\,dy$, first by direct calculation, and
second by using Green's theorem to relate this to an integral over a simpler
path.

\item Let $\Gamma$ be the curve in $\R^3$ which is the intersection of
the cylinder $x^2 + y^2 = 4$ and the surface $z = xy$, oriented
counterclockwise when viewed from above.  Compute the line integral
$\int_\Gamma (3y\,dx + yz\,dy + (e^{\sin z} - x^2/2)\,dz)$, first
by pulling back the integral to $\R^2$ and using Green's theorem, and
then by using Stoke's theorem directly (what is a suitable surface whose
(oriented) boundary is $\Gamma$, and how would you parametrize it?).

\item Consider the conical surface $S = \set{(x, y, z)\in \R^3 \mid
4(x^2+ z^2) = (2-y)^2, 0\le y\le 2}$.  Orient $S$ so
the normal vector has positive second coordinate at each point.
\begin{enumerate}
\item Find a parametrization for $S$ and $\partial S$.  Be careful about
orientations.
\item Find $\int_S d(x^3\,dx + y^3\,dy - x^3\,dz)$ in as many ways as
possible, including by the trick in the next part.  (A helpful formula:
$\cos^4\theta + \sin^4\theta = (3+\cos 4\theta)/4$.)
\item Find $\int_S y\,dx\,dy + (xz+y)dx\,dz + yz\,dy\,dz.$
\emph{Hint:} What is the integral over $S$ plus the flat disc with boundary $\partial S$?
\end{enumerate}

\item \begin{enumerate}
\item Show that the set $S$ of $n\times n$ real matrices
which are diagonalizable and have all eigenvalues different is an open
subset of the space ${\cal M}(n\times n)$ of all matrices.
\item Suppose a matrix $A$ satisfies $AB = BA$ for some $B \in S$.
Show that $A$ is diagonalizable.  Is $A$ in $S$?
\end{enumerate}

\item Let $f$ and $g$ be nonnegative
continuous functions on $I = [0,1]$, and suppose that
$f(x)g(x) \ge 1$ for all $x \in I$.  Show that $(\int_I f)(\int_Ig) \ge 1$.
\emph{Hint:} $\langle f, g\rangle = \int_0^1 fg$ defines an inner product on $\cC([0,1])$.

\item Put an inner product on ${\cal P}_n$ by
$ \langle f, g \rangle = \int_{-1}^1 fg$
(why is this positive definite?).  Consider the linear transformation
$T\colon {\cal P}_n \to {\cal P}_n$ defined by
\begin{equation*}
T(f) = \frac{d}{dx}\left(\left(1 - x^2\right)\frac{df}{dx}\right)
\end{equation*}
\begin{enumerate}
\item Show that $T$ is self-adjoint.  What are its eigenvalues?
\item Show that for any $k\le n$ there is a unique monic\footnote{Leading coefficient 1}
eigenvector of degree $k$; \ie of the form
\[p_k(x) = x^k + a_{k,k-1}x^{k-1} + \dots + a_{k,0}.\]
\item Show that the different $p_k$ are orthogonal. What is the eigenvalue of $p_k$?
\item What fails if we try to do the same thing with $T(p) = d^2p/dx^2$?
\end{enumerate}

\item
\begin{enumerate}
  \item Show that the limit $\lim_{t\to \infty} \int_1^t \frac{\sin x}{x}\,dx$
  exists (integrate by parts)

  \item Show that $\lim_{k\to \infty} \sum_{n=1}^k 1/n$ is infinite,
  by comparing the integral $\int_1^\infty dx/x$ with the integral of
  a step function on $[1, \infty)$.

  \item Show that the improper integral $\int_1^\infty \frac{\sin x}{x}$
  does not exist with the definition we have been using.  We say that the
  integral converges, but does not converge absolutely
  (\ie $\abs{\sin x/x}$ can't be integrated on $[1, \infty)$).

  \emph{Hint:} Show that
  \begin{equation*}
  \int_{2k\pi}^{2(k+1)\pi} \abs{\frac{\sin x}{x}} \ge \frac{2}{(2k+1)\pi}
  \end{equation*}
  and use (b).

  \item Prove the following related result about infinite series.
  Let $a_n$ be a \emph{decreasing} sequence of \emph{nonnegative} numbers,
  and suppose that $\lim_{n\to \infty} a_n = 0$. Show that
  \begin{equation*}
  \sum_{n=1}^\infty (-1)^na_n
  \end{equation*}
  converges, \ie the sequence of partial sums converges.

  \emph{Hint:} Define two sequences $e_n = \sum_{k=1}^{2n} (-1)^ka_k$ and
  $o_n = \sum_{k=1}^{2n-1} (-1)^ka_k$.  Show that $e_n$ is a
  decreasing sequence, $o_n$ is an increasing sequence, and both
  are bounded; thus they both have limits.  Look at $e_n - o_n$ and
  conclude that the limits are equal.
\end{enumerate}


\item
\begin{enumerate}
  \item Let $f\colon [0,1] \to \R$ be an integrable function, and let $P_n$
  be the partition of $[0,1]$ into $n$ equal parts: $P_n = \set{0,\frac{1}{n}, \frac{2}{n},
  \dots, 1}$.  Let $l_n = L(f, P_n)$ and $u_n = U(f, P_n)$, and
  $a_n = \frac{1}{n}\sum_{k=1}^n f(\frac{k}{n})$.  Show that all three of these sequences
  converge to $\int_{[0,1]}f$.

  \emph{Hint:} Given $\epsilon>0$, find a partition $P$ for which
  $U(f, P) - L(f, P)<\epsilon/2$.  Then find an $n$ so that $u_n - l_n$
  is close to $U(f, P) - L(f, P)$.

  \item Give an example of a nonintegrable function $f$ for which the
  three sequences converge but have different limits.

  \item Why didn't we define the integral to be the common limit of the sequences
  $\seq{l_n}$ and $\seq{u_n}$, when they are equal?
  Think about what would happen to the proofs of the basic properties of the integral.
\end{enumerate}

\item Let $T\colon \C^n\to \C^n$ be an invertible linear transformation
for which $T^2$ is diagonalizable.  Show that $T$ is diagonalizable.

\emph{Hint:} if $\vv$ is an eigenvector for $T^2$ but not for $T$, show that
there are two independent eigenvectors in the
span of $\vv$ and $\vec w = T\vv$.

\item (Matrix exponentiation) Let $M = {\cal M}(n\times n)$ be the vector space
of all $n\times n$ real matrices.  Define a function $\exp\colon M\to M$ by
\begin{equation*}
\exp(A) = \Id + A + \frac{A^2}{2} + \dots + \frac{A^n}{n!} + \dotsb
\end{equation*}
\ie the limit of the sequence of partial sums.  Show that this converges
for any $A$ (you may find it useful to know that there is a norm on $M$ for which
$\lnorm AB \rnorm \le \lnorm A \rnorm \lnorm B \rnorm$).  For $n=1$ this agrees with
the usual exponential function $\exp(a) = e^a$.
\begin{enumerate}
  \item If $B$ is an invertible matrix, show that $\exp(BAB^{-1}) = B\exp(A)B^{-1}$.
  \item If $A$ is diagonalizable, show that $\det (\exp A) = e^{\tr A}$.
  This is actually true for any matrix.
  \item Calculate
  $\exp \left( \begin{smallmatrix} 1 & a \\ 0 & 1 \end{smallmatrix} \right)$.

  \item Suppose $A$ and $B$ are diagonalizable, and $AB = BA$.  Show
  that $\exp(A+B) = \exp(A)\exp(B)$.  Find diagonalizable matrices $A$ and $B$
  for which the formula fails.

  \item If $A$ is a skew-symmetric matrix\footnote{$A = -A^t$}, show that
  $\exp(A)$ is orthogonal.

\end{enumerate}

\end{enumerate}
\end{document}
