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\begin{center}
{Mathematical Physics --- PHZ 3113}\\\vspace {2mm} 
{Midterm~1 (February 18, 2013)}  % \\
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\centerline{------------------------------------------------------}

\bigskip

\begin{enumerate}

\item Calculate the gradient of the 3D potential (in arbitrary units) 
$\frac{1}{2}\,r^2$.

\item A force in 3D is (in arbitrary units) given by $\vec{F} =
-r\,\vec{r}$. Use (with Einstein convention)
\begin{eqnarray} \nonumber
  \nabla\times r\,\vec{r}\ =\ \epsilon_{ijk}\,\hat{x}_i\,\partial_j\,r\,x_k
\end{eqnarray} 
to calculate the curl of this force.

\item Calculate the curl of the force
\begin{eqnarray} \nonumber
  \vec{F} = x_2\,\hat{x}_1+x_3\,\hat{x}_2+x_1\,\hat{x}_3 
          = \left(\matrix{x_2\cr x_3\cr x_1}\right)\,.
\end{eqnarray} 

\vfil\newpage

\item A point mass on an inclined plane experiences under gravity
a force $\vec{F}=-F\,\hat{z}$ in downward direction (see the figure).
Find the magnitudes of the forces perpendicular and parallel to the
plane as function of the angle $\alpha$.

\item A point mass is suspended as shown in the figure. As in the
previous problem it experiences a force $\vec{F}=-F\,\hat{z}$ in 
downward direction. Find the $z$ components of the tensions 
$\vec{T}^{\,1}$ and $\vec{T}^{\,2}$ as functions of the angles 
$\alpha_1$ and $\alpha_2$. In which limit becomes $T^1_x$ infinite?

How many free parameters and how many equations are there? Write
down the equations.

\end{enumerate}

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