Line integral: $u = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}, z)$ Let $u = ( \frac{-y}{x^2 + y^2}, \frac{x}{x^2 + y^2}, z)$ and D the domain bounded by the torus obtained by rotating the circunference $(x-2)^2 + z^2 =1, y=0$ around the z-axis. Show that $rot( u )=0 $ in D, but 
$$ \int_C u.dr \neq 0$$
if $C$ is the circunference $x^2 + y^2 = 4, z=0$. Determine all the possible values for the integral 
$$\int_{(2,0,0)}^{(0,2,0)} u.dr$$
over a path in $D$
My attempt:
I've show the first part but I am having troubles with the last one. If $y$ is not zero, then $\phi(x,y,z)=\arctan(-\frac{x}{y})+\frac{z^2}{2}$ would be a good potential for $u$, but I couldnt find the case that $y=0$. 
Thanks!
 A: Answer:
parameterization of C  is $$x = 2cost, y = 2 sint$$
$$r = 2costi + 2sint j + 0k$$
$$r' = -2sinti +2cost j$$
$$u = <-2sint/4, 2cost/4,0> = <-sint/2, cost/2,0>$$
$$u.dr =  (cos^{2}t+sin^{2}t) = 1$$
(0,2,0) in the curve translates to $\pi/2$ and (2,0,0) translates to 0
Thus the given line integral can then be written as
$$\int_{0}^{\pi/2} 1 dt$$
$$ =  t |\pi/2,0$$
$$ = \frac{\pi}{2}$$
It has been a long time since I have done this kind of problem, hopefully it is correct.
Thanks
Satish
A: This is rather belated, but the formula $\arctan(\frac{y}{x}) + \frac{1}{2}z^{2}$ for the potential of $u$ is correct only for $x > 0$ (assuming the potential is continuous and takes the value $0$ along the positive $x$-axis).
A "formal" potential for $u$ (away from the $z$-axis) is the multi-valued function $\theta + \frac{1}{2}z^{2}$, with $\theta$ the "longitude" in cylindrical coordinates.
Since $\theta(0, 2, 0) = \frac{\pi}{2} + 2\pi k$ and $\theta(2, 0, 0) = 2\pi\ell$ for some integers $k$ and $\ell$, the line integral of $u$ from $(2, 0, 0)$ to $(0, 2, 0)$ along a path in $D$ can take any value $\frac{\pi}{2} + 2\pi k$ with $k$ an integer.
