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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 7}
\smallskip

\end{center}

%\comments{ 
{\sc Due:} Thurs. Oct.~29 in class. [Late papers will be
accepted (without penalty) until 1:00 PM Friday.] %} \bigskip
\bigskip

\centerline{\fbox{
{\bf Note:} The date of Exam 2 has been changed to Tuesday, Nov. 10.}}
\bigskip

 Please carefully read Sections 8.1--8.4 in the Marsden-Hoffman text. 
\medskip

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.~413 \#1, 2, 3,\qquad p. 444 \#35 

\bigskip

\begin{problems}

\item Let $f:\R^2 \to \R$ be a smooth map.
\begin{parts}
\item If $\norm{\nabla f(x)} \le M$ everywhere, show that $\norm{f(x)
- f(y)} \le M\norm{x-y}$.
\item Let $A$ be the annular region $A:=\{x\in \R^2\,:\, 1\le \norm{x}
  \le 2\}$ and $f:A \to \R$ a smooth map.  If $\norm{\nabla f(x)} \le
  M$ for all points in $A$, estimate $\norm{f(x) - f(y)}$ for $x$ and
  $y$ in $A$.
\end{parts}

\item {\sc Proof or  Counterexample?}  There is {\it no} smooth function defined on $\R^2$ with exactly two critical points, both non-degenerate local minima.

\item If $h(x,y)= x^2 -2xy +5y^2$, since then
$h(x,y)= (x-y)^2 + 4y^2$, it is clear that under the change of
coordinates $u=x-y$, $v=2y$ we can write $h=u^2+v^2$ as a sum of
squares.  

Prove (with your bare hands, {\it not} using the Morse Lemma) that one 
can do this near the origin for any smooth function $f(x,y):\R^2\to \R$ 
with the properties that
$$
f(0,0)=0, \qquad  f'(0,0)=0, \qquad  f''(0,0) \text{ is positive
  definite}.
$$
[Here $f'$ is the gradient and $f''$ the second derivative matrix.]

\item
\begin{parts}
\item Let $f \colon \R \to \R$ be a smooth function with the properties
that $f''(x)\ge 0$ and $f(x)\le C$ for all $x \in \R$.  Show
that $f(x) = \text{constant}$.
\item Let $f \colon \R^2 \to \R$ be a smooth function with the
properties that the hessian matrix $f''(x)$ is positive semi-definite and
that $f(x)\le C$ for all $x \in \R^2$.  Does this imply that
$f(x)= \text{constant}$? Proof or counterexample.
\end{parts}


\item{} [Marsden-Hoffman, p.~439 \#6]  Determine whether the ``curve'' described by the equation $x^2 + y +\sin(xy)=0$ can be written in the form $y=f(x)$ in a neighborhood of $(0,0)$.

Does the implicit function theorem allow you to say weather the equation can be written in the form $x=h(y)$ in some neighborhood of $(0,0)$?

\item Let $f:\R^2 \to \R^2$ with $f(x,y)=(u(x,y),\,v(x,y))$ and assume that $u$ and $v$ satisfy the \emph{Cauchy-Riemann equations}
$$
u_x(x,y)=v_y(x,y)\qquad\text{and}\qquad u_y(x,y)=-v_x(x,y).
$$
\vskip -20pt
\begin{parts}
\item Show that this map is invertible near a point $(x,y)$ if and only if $Df(x,y) \ne 0$.
\item  Show that the inverse map also satisfies the Cauchy-Riemann equations.
\end{parts}

\item{} [Marsden-Hoffman p.~420 \#4] Find the extrema of $f(x,y,z)=x+y+z$ subject to the constraints: $x^2+y^2=1$, $2x+z=1$.

\item{} [Marsden-Hoffman p.~444 \#35] Find the relative extrema of $f(x,y)=x^2+y^2$ subject to the constraint $x^2-y^2=1$.

\item{} [Marsden-Hoffman p.~444 \#38] A rectangular box with no top is to have a surface area of $16$ square meters.  Find the dimensions that maximize the volume.


\item Let $A$ be a real square symmetric matrix and let $v\in\R^n$
be a point on the unit sphere, $\norm{x}=1$, where $f(x)=\ip{x}{Ax}$ has its maximum.  Show that $v$ is an eigenvector of $A$.  What is the corresponding eigenvalue?

\item {}{\sc [continuation of the previous problem]}  Say $w\in \R^n$ is a point maximizing $f(x)$ in the set $\norm{x}=1$ with $w$ also perpendicular to $v$, so $\ip{w}{v}=0$.  Show that $w$ is also an eigenvector of $A$.
\end{problems}

%\comments{
\vskip 20pt

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

\begin{problems}

\item [B-1]
\begin{parts}
\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}

 \end{problems}
%}




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