\documentclass[12pt]{article} % BB Apr 06 2009.
\usepackage{epsfig}
\textheight 9.0in \textwidth  6.0in \topmargin -0.4in \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{$\underline{\smash{\hbox{#1}}}$}
\pagestyle{empty}

\begin{document}

\begin{center}
{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 15}\\%
\vspace {3mm} (April 20, 2009) \\
\end{center}
\setcounter{equation}{0}

\begin{description}

\item{(62)} \vskip -60pt
\begin{enumerate}
\item Write $\left(\vec{a}\times\vec{b}\,\right)^k$ using $\epsilon^{klm
      }$ and the summation convention.  Due April~17 in class (1~point).
\item Write $\left[\nabla\times\left(\vec{a}\times\vec{b}\,\right)
      \right]^i$ using $\epsilon^{ijk}\partial_j$ and the summation 
      convention. Due April~17 in class (1~point).
\item Perform in the last expression the sum over $k$.
      Due April~17 in class (1~point).
\item Show that the last expression agrees with the $i$ component of
      $$  \left(\nabla\cdot\vec{b}\right)\,\vec{a}
        + \left(\vec{b}\cdot\nabla\right)\,\vec{a}
        - \left(\nabla\cdot\vec{a}\right)\,\vec{b}
        - \left(\vec{a}\cdot\nabla\right)\,\vec{b}\,. $$
      Due April~20 in class (2~points).
\end{enumerate}
 

\item{(63)} Apply the vector identity of the previous assignment to
$\nabla\times\left(\vec{B}\times\vec{v}\,\right)$. Discuss which terms
on the r.h.s.\ are zero and omit them. Due April~20 in class (3~points).
 

\item{(64)} Consider the differential equation
$$ L\frac{d^2Q}{dt^2} + R\frac{dQ}{dt}+\frac{Q}{C} =
   \epsilon_{\max} e^{-i\omega t}\ .$$
\begin{enumerate}
\item Determine $Q_0$ for the special inhomogeneous solution $Q_i=Q_0
      \,\exp(-i\omega t)$. Due April~22 in class (2~points).
\item Find the general homogeneous solution from the ansatz $Q_h\sim
      \exp(i\alpha t)$. Due April~22 in class (4~points).
\end{enumerate}

\end{description}

% \begin{figure}[tb] \vspace{5pc}
% \centerline{\hbox{ \psfig{figure=figs/kepler.eps,width=17.5cm} }}
% \end{figure}

\end{document}

