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\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
Curl Homework (January 30, 2013) \\
\end{center}
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\noindent Use the identity (with Einstein convention)
\begin{eqnarray} \label{epsdel}
  \epsilon_{ijk}\epsilon_{klm} =
  \delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}
\end{eqnarray} 
to solve the following assignments.

\begin{enumerate}
\item Show (Book (1.89) p.51)
\begin{eqnarray} \nonumber
  \vec{B}\times\left(\nabla\times\vec{A}\right) +
  \vec{A}\times\left(\nabla\times\vec{B}\right)&=&\\ 
  \nabla\,\left(\vec{A}\cdot\vec{B}\right) -
  \left(\vec{B}\cdot\nabla\right)\,\vec{A} -
  \left(\vec{A}\cdot\nabla\right)\,\vec{B}&.&~~~
\end{eqnarray} 

\item The vector potential $\vec{A}$ of a magnetic dipole moment
$\vec{m}$ is given by (Book 1.7.11 p.52)
\begin{eqnarray} \nonumber
  \vec{A} = \left(\frac{\mu_0}{4\pi}\right)\,
 \left(\frac{\vec{m}\times\vec{r}}{r^3}\right)\,.
\end{eqnarray} 
Calculate the magnetic field $\vec{B}=\nabla\times\vec{A}$.

\end{enumerate}

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