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\begin{document}

\title{Analysis II}
\author{Homework 12}
\date{Due in class Monday, May 13, 2019}
\maketitle

\begin{enumerate}

\item (Kantorovitz, Exercise 4.2.15.1) Let $D$ be the domain bounded
by the parabola $x=y^2$ and the line $x=y$.  Calculate the integral
\[
  \int_D \sin \frac{\pi x}y \, \d S \,.
\]

\item (Kantorovitz, Exercise 4.2.15.2) Let $D$ be the annulus bounded
by circles centered at the origin with radii $0<a<b$.  Compute
\[
  \int_D \arctan \frac{y}x \, \d S \,.
\]

\item (Kantorovitz, Exercise 4.2.15.6) Find the area of the closed
domain bounded by the two parabolas $y^2 = a x$ and $y^2 = b x$
($0<a<b$), and the two hyperbolas $xy = \alpha$ and $xy = \beta$
($0<\alpha<\beta$).

\emph{Hint:} Use the map
\[
  (x,y) \mapsto (u,v) \equiv (y^2/x,xy) \,.
\]

\item Let $D$ be a domain in $\R^n$.  A function $f \in C^2(D)$ is
called \emph{harmonic} if $\Delta f=0$, where $\Delta$ is the
\emph{Laplace operator} defined via
\[
  \Delta f = \nabla \cdot \nabla f = \partial_1^2 f + \dots +
  \partial_n^2 f \,.
\]
Show that a harmonic function has the mean value property
\[
  f(x) = \frac1{S(\partial B(x,r))}
  \int_{\partial B(x,r)} f \, \d S
\]
for every $x \in D$ and every $r>0$ sufficiently small such that
$\partial B(x,r) \subset D$, where $B(x,r)$ denotes the ball centered
at $x$ with radius $r$ and
\[
  S(\partial B(x,r)) = \int_{\partial B(x,r)} \d S
\]
is the $(n-1)$-dimensional content of $\partial B(x,r)$.

\emph{Hint:} Proceed in the following steps:
\begin{enumerate}
\item Define
\[
  \phi(r) = \dfrac{\displaystyle \int_{\partial B(x,r)} f \, \d S}%
                  {\displaystyle \int_{\partial B(x,r)} \d S} \,.
\]
Now use a change of variables in numerator and denominator which takes
$B(x,r)$ to $B(0,1)$.

\item Differentiate with respect to $r$, move the differentiation
under the integral, and apply the chain rule.

\item Apply the divergence theorem and use that $f$ is harmonic to
conclude that $\phi'(r) = 0$.

\item Finish the proof by considering the limit $r \to 0$.
  
\end{enumerate}


\end{enumerate}

\end{document}
