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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 9}
\smallskip

\end{center}

%\comments{ 
{\sc Due:} Thurs. Nov.~19 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.~466 \#3, \ p.~469 \#2, \ p,~490 \#5, \ p.~505 \#4, \ p.~508 \#3, \
p.~508 \#5, \ p.~512 \#5 
%}
\bigskip

\begin{problems}

\item In the plane $\R^2$, prove that the points on the graph of the curve $y=x^2$ have measure zero.

\item Determine if the following improper integrals converge or converge absolutely.  Justify your assertions.

a) \ $\displaystyle\int_1^\infty \frac{\sin x}{1+ x}\,dx$ \hskip 4.0cm 
b) \ $\displaystyle\int_1^\infty \frac{\sin x}{1+ x^2}\,dx$


\item Let $f:\R\to\R$ be a continuous function with $f(x)\ge 0$.  If the improper integral $\int_0^\infty f(x)\,dx$ exists, then $\lim_{x\to\infty} f(x)=0$.  Proof or counterexample.

\item {} [Marsden-Hoffman, p.~ 493 \#35] Let $A_n=[(n+1) +(n+2)+\cdots+(n+n)]/n$.  Prove that $\lim_{n\to\infty}A_n/n=3/2$.  [{\sc Suggestion: } Use the Riemann integral.]

\item {} [Marsden-Hoffman, p.~494 \#41] Let 
$\Cal{R} = \{ f:[0,1] \to \R\ |\  f \ \text{is Riemann Integrable\ } \}$ 
and define
$$
d(f,\,g) = \int_0^1 \abs{f(x)-g(x)}\,dx.
$$
Is $d$ a metric on the space $\cal{R}$?  Why?

\item{} [Marsden-Hoffman p.~495 \#47] For every $\alpha>0$, compare $\int_0^N x^\alpha\,dx$ with $\sum_{n=1}^{N} n^\alpha$ and  $\sum_{n=0}^{N-1} n^\alpha$
and hence determine
$$
\lim_{N\to\infty}\frac{1}{N^{1+\alpha}}\sum_{n=1}^Nn^\alpha.
$$

\item Let $f: [a,\,b]\to \R]$ be an invertible continuous function,
  $y=f(x)$ and let $\int_a^b f(x)\,dx = F(b)-F(a)$, so $F'=f$. Say
  $x=\varphi(y)$ is the inverse function of $f$. Let $\alpha=F(a)$ and
  $\beta=F(b)$.
\begin{parts}
\item Show that 
$$
\int_\alpha^\beta \varphi(y)\,dy +\int_a^bf(x)\,dx =bf(b)-af(a).
$$
Equivalently,
$$
\int_\alpha^\beta \varphi(y)\,dy =bf(b)-af(a) -[F(b)=F(a)].
$$
Thus, if you can explicitly integrate a function, you can also
integrate the inverse function.
\item Apply this to the examples $y= x^2$ and $y=e^x$.
\end{parts}

\item Let $f(x)\in C([a,b])$.  Show that
$$
\exp\left[\frac{1}{b-a}\int_a^b f(x)\,dx\right] \le
    \frac{1}{b-a}\int_a^b \exp [f(x)]\,dx
$$
Similarly, let $\Cal{D} \in \R^2$ be a bounded region whose boundary
is smooth enough that the integrals below all exist.  Define the {\it
average}, $\bar h$, of a continuous function $h$ by $\bar h =
\frac{1}{\text{Area}(\Cal{D})}\iint_{\Cal{D}} h \,dA$, where $dA$ is
the usual element of area.  Show that
$$
\frac{1}{\text{Area}(\Cal{D})}\iint_{\Cal{D}} e^h \,dA \ge e^{\bar h}
$$

\item If $A$ is a positive definite $2\times 2$ matrix and $b\in\R^2$ is any vector, show that the improper integral 
$$
\iint_{R^2} e^{-[\ip{x}{Ax} +\ip{b}{x}]}\,dx\,dy
$$
exists.  [Challenge: find an explicit formula for this in terma of $A$
and $b$≈ß.]

\item
\begin{parts}
\item Compute
$$
\iint_{\R^2}\frac{dx\,dy}{(1 + 4x^2 + 9y^2)^2},
\
\iint_{\R^2}\frac{dx\,dy}{(1 + x^2 + 2xy + 5y^2)^2},
\
\iint_{\R^2}\frac{dx\,dy}{(1 + 5x^2 -4xy+ 5y^2)^2}.
$$

\item Compute $\displaystyle
\iint_{\R^2}\frac{dx_1\,dx_2}{[1 + \ip{x}{Cx}]^2}$, where
$C$ is a positive definite (symmetric) $2 \times 2$ matrix, and
$x = (x_1,x_2) \in \R^2$.
\end{parts}

\item {} [{\sc Generalization of the previous problem}]
Let $h(t)$ be a given function and say you know that $\int
_0^\infty h(t)\,dt = \alpha$.  
\begin{parts}
\item If $C$ be a positive definite $2 \times
2$ matrix.  Show that
$$
\iint_{\R^2} h(\,\ip{x}{Cx}\,)\ dA = \frac{\pi\alpha}{\sqrt{\det C}}.
$$
\item Generalize this to obtain a formula for
$$
\iint_{\R^n} h(\,\ip{x}{Cx}\,)\ dV,
$$
where now $C$ be a positive definite $n \times n$ matrix.  The answer
will involve some integral involving $h$ and also the ``area" of the
unit sphere $S^{n-1}$ in $\R^n$.
\end{parts}

\item a) \ Compute
$\displaystyle \iint_{\kern -0pt \R^2} e^{-(5x^2 -4xy + 5y^2)}\,dx\,dy$.
\hskip 1cm b) \ Compute $\displaystyle
    \iint_{\kern -0pt \R^2} e^{-(5x^2 -4xy + 5y^2 -2x +3)}\,dx\,dy$.


\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}
