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

\begin{center}
{\large\bf Problem Set 8}
\smallskip

\end{center}

%\comments{ 
{\sc Due:} Thurs. Nov.~5 in class. [Late papers will be
accepted (without penalty) until 1:00 PM Friday.] %} \bigskip
\bigskip

\centerline{\fbox{
{\bf Note:}  Exam 2 is on Tuesday, Nov. 10.}}
\bigskip

 Please carefully read Sections 8.1--8.5 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.~413 \#1, 2, 3,\qquad p. 444 \#35 
}
\bigskip

\begin{problems}

\item Compute $\displaystyle\int_0^c x^3\,dx$ with your bare hands by using a Riemann sum, partitioning the interval $0\le x \le c$ into sub-intervals of equal length.  You may use without proof that
$$
1^3+2^3+3^3+\cdots+n^3 = \left[\frac{n(n+1)}{2}\right]^2.
$$
\vspace{-20pt}
\item Compute $\displaystyle\int_0^b \sin x\,dx$ by using a Riemann sum, partitioning the interval $0\le x \le b$ into sub-intervals of equal length.  You may use without proof that
$$
\sin x + \sin 2x+\cdots+\sin nx = 
\frac{\cos\frac{x}{2} - \cos(n+\frac{1}{2})x}{2\sin\frac{x}{2}}.
$$
[One way to get this is by summing the geometric series
$e^{ix} + e^{i2x}+\cdots +e^{inx}$ and taking the imaginary part].

\item {} [Marsden-Hoffman, p.~456 \#2--3] On the set $A=[0,1]\times[0,1]\subset\R^2$, let $f(x,y)=1$ for all $(x,y)\in A, \ x\ne 0$ and $f(0,y)=0$.  Prove that $f$ is Riemann integrable on this set and compute its value. 

\item{} [Marsden-Hoffman, p.~454 \#1]. Show that the ``volume'' of the set $\{(x,y)\in\R^2\,|\, x^2+y^2=1\}$ is zero.

\item{} [Marsden-Hoffman, p.~454 \#2]. Show that the three dimensional measure of the $xy$ plane in $\R^3$ is zero.

\item{} [Marsden-Hoffman, p.~454 \#5]. Must the boundary of a set of measure zero have measure zero?  Proof or counterexample.


\item Let $f$ be continuous in the square $ Q:=\{[0,\,1]\times[0,\,1]\subset\R^2\}$ and
assume that $f(x,y)\ge 0$ for all points in $Q$.  Use the definition of the integral as a Riemann sum to show that if $\int_Q f(x,y)\,dx\,dy =0$,
then $f(x,y))=0$ everywhere.  [You will need to use that since $f$ is
continuous, if it is positive at some point, then it is positive in
some rectangle containing the point.]
\smallskip

This may be false if $f$ is discontinuous somewhere. Example?


\item Let $T:\R^2 \to \R^2$ be a smooth map $T:(u,v)\mapsto (x(u,v),\,y(u,v))$.  Assume that it is an invertible map from the set $A$ to the set $B$, and that $DT(u,v)$ is invertible at every point of $A$.  Let $f(x,y)$ be a smooth function for all points $(x,y)\in B$ and let
$$
g(u,v)=  f(x(u,v),\,y(u,v))
$$
(perhaps think of $T$ as a smooth change of coordinates).
\begin{parts}
\item Show that the point $x=p$, $y=q$ in $B$ is a critical point of a (smooth) function $f(x,y)$ if and only if $(p, q)$ is the image of critical point of 
$(a,\,b)$ of $g(u,v)$.
\item Say $(a,\,b)$ is a \emph{non degenerate} critical point of $g(u,v)$ so $p=x(a,b)$, $q==y(a,b)$ is a critical point of $f$.  Show that $(p,\,q)$ is a non-degenerate critical point of $f$ and that it is a local minimum if and only if $(a,b)$ is a local minimum of $g$. [If it helps, you may assume that the Hessian $g''(a,\,b)$ is a diagonal matrix.]

[{\sc Moral: } if you think of the map $T$ as a change of coordinates, this says that $T$ preserves both the critical points of $g$ and if they are local maxima, minima, or saddles.]
\end{parts}

\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}
