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\begin{document} \huge

\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
{\bf Gradient,} (January 23, 2013) \\
\end{center}
\setcounter{equation}{0}

\rightline{Group \#~~~~~~~~}

\leftline{Participating students (print):} \vskip 60pt

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

\bigskip

\begin{enumerate}

\item Calculate
\begin{eqnarray}
  \frac{\partial~}{\partial x_i}\,x_j\,.
\end{eqnarray} 

\item Calculate
\begin{eqnarray}
  \frac{\partial~}{\partial x_i}\,\sum_{j=1}^n x_j^2\,.
\end{eqnarray} 

\item Calculate
\begin{eqnarray}
  \frac{\partial~}{\partial x_i}\,\sqrt{\sum_{j=1}^n x_j^2}\,.
\end{eqnarray} 

\item Calculate
\begin{eqnarray}
  \nabla\,r\,.
\end{eqnarray} 

\item Calculate
\begin{eqnarray}
  \nabla\,f(r)\,.
\end{eqnarray} 

\item Example: Calculate the electric field $\vec{E}$ from a
given central potential:
\begin{eqnarray}
  \vec{E}=-\nabla\,\Phi(r)\,,~~\Phi(r)=\frac{q}{4\pi\,\epsilon_0\,r}\,.
\end{eqnarray} 

\item Calculate
\begin{eqnarray}
  \nabla\,\left(\vec{n}\cdot\vec{r}\right)
\end{eqnarray} 
where $\vec{n}$ is a constant vector. Describe the solutions $\vec{r}$
for $\vec{n}\cdot\vec{r}=0$ in words.

\end{enumerate}

\end{document}

