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\title{Midterm exam ~~ Math 5710 ~ Fall 2013}
\author{Your name....................}
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1. The electrical circuit is given in Figure 1. Find the current between nodes $A$ and $B$. Show algebraic formulas.

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2. Show, computing {\em curl\, v}, that 

(a) $v_1=(y^2, x)$ is not the gradient of any function, but 
$$(b) \qquad v_2= \left(\frac{2x}{y}, -\frac{x^2}{y^2} \right) $$
is. Find the potential. 


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3.  Can  vector field $ v_3=( xy, x)$ be represented as {$v_3$ =\em curl\,w} where $w$ is some vector field? 

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4.  The force field $F$ depends on the coordinates $(x, y, z)$ as $F=
(y^2 z, 2 x\,y\, z, ?)$. What is the third component of the force, if
it is known  that $F$ is conservative $F=\nabla u$. 

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5.  In pure rotation, the velocity field $v$ is represented as $ v=(\omega_1, \omega_2,\omega_3) \times (x, y, z)$. Compute {\em curl}\,$v$, {\em div}\,$v$. Show that $v$ and $\omega$ are perpendicular. 

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6. Compute 
$\nabla \cdot (u \times v) $, where $u, v$ are two vectors.
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