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%==================== END PREAMBLE ==============
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\vspace*{-50pt}
{\bf  Math 361, Fall 2015\hfill Jerry L. Kazdan}
\medskip

\begin{center}
{\large\bf Problem Set 10}
\smallskip

\end{center}

%\comments{ 
{\sc Due:} Thurs. Dec.~3 in class. [Late papers will be
accepted (without penalty) until 1:00 PM Friday.] %} \bigskip
\bigskip

 Please carefully read Chapter 9 in the Marsden-Hoffman text. 
%\medskip

%\comments{
The following short answer problems from Marsden-Hoffman are not
assigned, but you should know how to do them.  Some short answer
problems will be on our exams.

\medskip
p.~508 \#5, \ p.~512 \#4, \ p.~539 \#24 
%}
\bigskip

\begin{problems}

\item Compute the arc length of the spiral helix $\gamma(t)=(t\cos
  t,\,t\sin t, t)$ for $0\le t \le 2\pi$.

\item Consider the curve $y=x\sin(1/x)$ for $0<x\le 1$ while $y=0$
  when $x=0$.  Show that this curve is not rectifiable (has infinite
  length).

\item Let $A$ be a real positive definite $n\times n$ matrix and
  $c>0$.  Compute the ``volume'' of the set of points in $x \in \R^n$
  where $\ip{x}{Ax} \le c$.

\item Let $A$ be a real positive definite $n\times n$ matrix, $b$ a
vector in $\R^n$ and $c$ a real number.  Fow which value(s) of $c$
is the set of points in $x \in \R^n$ where
$$
\ip{x}{Ax} +2\ip{b}{x} +c =  0
$$ 
not empty?  [{\sc Suggestion: } First answer this in the special case
$n=1$.].

\item Let $\gamma(s)$ define a smooth curve parametrized by arc length
  $s$, so $\norm{\gamma'(s)}=1$.  Show that the vector $\gamma''(s)$
  is perpendicular to the tangent vector $\gamma'(s)$.

\item Let $0<b<a$.  In class we parametrized the standard torus
  (surface of a doughnut) in $\R^3$ as
$$x=(a+b\cos\phi)\cos\theta, \qquad  y=(a+b\cos\phi)\sin\theta, \qquad  z=b\sin\phi.
$$
Show this is a smooth two dim\textsf{}ensional surface and compute the equation
of the tangent plane at the point \ $\theta=\pi/2$, \ $\phi=0$.

\end{problems}

%\newpage

%\comments{
\vskip 20pt

\begin{center}{\large \bf Bonus Problems}
            
[Please give these directly to Professor Kazdan]
\end{center}

\begin{problems}

\item [B-1]
\begin{parts}
\item Let $V=(v_1,\ldots,v_n)\ne 0$ be a real vector and $A$ be the
  $n\times n$ matrix $A=(v_iv_j)$, so the product $v_iv_j$ is the $ij$
  element of $A$.  Find the eigenvalues of $A$. [Note: $A$ has rank
  one since each column is a multiple of $V$.  Thus
  $\lambda_1=\lambda_2=\cdots=\lambda_{n-1}=0$.  What is the trace of
  $A$? What is $\lambda_n$?]

  Let $B=A+cI$, where $c\in\R$.  Compute the eigenvalues and the
  determinant of $B$.
\item If $u(x_1,x_2,\ldots,x_n)$ is a given smooth function, let $u''
  := \left(\frac{\partial^2u} {\partial x_i \partial x_j}\right)$ be
  its second derivative (Hessian) matrix.  Find all solutions of $\det
  (u'') = 1$ in the special case where $u=u(r)$ depends only on $r =
  \sqrt{x_1^2 + \cdots x_n^2}$, the distance to the origin.
\item Let $x = (x_1,x_2,\ldots,x_n)$ and $A$ be a square matrix with
  $\det A =1$.  If $u(x)$ satisfies $\det (u'') = 1$ (see above), and
  $v(x) := u(Ax)$, show that $\det(v'') = 1$ also. [Remark: the
  differential operator $\det (u'')$ is interesting because its
  symmetry group is so large.]
\end{parts}

\item [B-2]\begin{parts}
  \item On the real number line, consider two points $x$ and $\hat{x}$
    to be equivalent if $x-\hat{x}=k$ for an integer $k$.  If $c$ is
    an irrational number, show that the points $nc$, $n=1,2,\ldots$
    are all distinct and are dense in the interval $[0,1]$.

  \item The \emph{flat torus} in $\R^2$ is the set of points $x,\,y$
    where we consider the point $\hat{x},\,\hat{y}$ to be equivalent
    to $(x,\,y)$ if $x-\hat{x}$ and $y-\hat{y}$ differ by
    integers. (possibly different integers).

    Consider the line $y=cx$ in the plane where $c$ is an irrational
    number.  Show that viewing this line as on the flat torus, it does
    not intersect itself and its points are dense in the torus.
    [Suggestion: First understand the simpler case where $c$ is a
    rational number and show that you get a closed curve.]
\end{parts}

 \end{problems}
%}




\medskip
[Last revised: \today]
\end{document}
