\documentclass[12pt]{article} % BB Mar 10 2013.
\textheight 9.0in
\textwidth  6.0in
\topmargin -1.0in  
\headsep    0.4in  
\oddsidemargin  0.25in  
\evensidemargin 0.25in  

\def\sqr#1#2{{\vcenter{\hrule height.#2pt
   \hbox{\vrule width.#2pt height#1pt \kern#1pt
      \vrule width.#2pt}
   \hrule height.#2pt}}}
\def\square{{\mathchoice\sqr64\sqr64\sqr{4.2}3\sqr{3.0}3}}
\def\undertext#1{$\underline{\smash{\hbox{#1}}}$}
\usepackage{bigints}
% \pagestyle{empty}

\begin{document} \Large

\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
{Midterm~2 (March 18, 2013)}  % \\
\end{center}
\setcounter{equation}{0}

\centerline{------------------------------------------------------}

\begin{enumerate}

\item Use the definition $\nabla\times\vec{A}=\epsilon_{ijk}\hat{x}_i
\partial_jA_k$ (with Einstein convention) and properties of the 
Levi-Civita tensor $\epsilon_{ijk}$ to transform
\noindent 
\begin{eqnarray} \nonumber
  \nabla\times\nabla\times\vec{A}
\end{eqnarray} 
into applications of the $\nabla$ operator, which do no longer
involve the curl. % \bigskip

\item Calculate the volume of an ellipsoid
$$ V\ =\ \int_{\left(\frac{x_1}{a}\right)^2+\left(\frac{x_2}{b}\right)^2
              +\left(\frac{x_3}{c}\right)^2 \le 1} d^3x\,.            $$
Hint: With suitable substitutions the integral can be
mapped on the integral for the volume of a sphere. %  \bigskip

\item Calculate $$\vec{v}=\vec{\omega}\times\vec{r}$$ and express the
result in spherical coordinates. Here $\vec{\omega}=\dot{\phi}\,\hat{z}$
is the angular velocity and $\vec{r}=r\,\hat{r}$ the position vector.
% \bigskip

\item Calculate the time derivative
$$ \dot{\hat{r}}\ =\ \frac{d\,\hat{r}}{d\,t}$$
of the unit vector
$$ \hat{r}\ =\ \frac{\vec{r}}{\left|\vec{r}\right|} $$
and express the result in spherical coordinates. % \bigskip

\item The orbit of a planet is assumed to be an ellipse, which in 
cylindrical coordinates is given by the equation
\begin{eqnarray} \nonumber
  \rho\ =\ \frac{p}{1+\epsilon\,\cos\phi}
\end{eqnarray} 
where $0\le\epsilon<1$ is the eccentricity and $p$ the latus rectum.
From the analytical solution of Kepler's problem it follows that
the magnitude of the angular momentum is (up to a correction for 
the finite sun mass) given by
$$ L={\rm const}\,m\,\sqrt{p}\,, $$
where $m$ is the mass of the planet. Use this information and angular 
momentum conservation to derive Kepler's 3rd law. %  \bigskip
\vfil

\end{enumerate}

\end{document}
