I need to calculate curl of $F$, and show that it is conservative on this region.
A vector field $F$, defined on a simply-connected region $r > 0,\; \frac{\pi}{4}< \theta < \frac{3 \pi}{4},\; \frac{- \pi}{2}< \phi < \frac{ \pi}{2}$ is expressed in spherical coordinates as:
$$F(r, \theta, \epsilon) \;=\; \sin(\phi) \cos(\theta) e_r - \sin(\phi)\sin(\theta) e_\theta + \cos(\phi)cot(\theta) e_\phi.$$
Curl in spherical coordinates:
$$\nabla * F \;=\; (\frac{1}{r}\frac{\partial F \phi}{\partial \theta} - \frac{1}{r\sin(\theta)}\frac{\partial F \theta}{\partial \phi} + \frac{\cot(\theta)}{r}F \phi)e_r \\ + (-\frac{\partial F \phi}{\partial r} - \frac{1}{r\sin(\theta)}\frac{\partial F r}{\partial \phi} - \frac{1}{r}F \phi)e_\theta \\ + (\frac{\partial F \theta}{\partial r} - \frac{1}{r}\frac{\partial F r}{\partial \theta} + \frac{1}{r}F \theta)e_\phi.$$
My workings using the above:
$\nabla * F$ = $(-\frac{\csc^2(\theta)\cos(\phi)}{r}-\frac{\cos(\phi)}{r}+\frac{\cos(\phi)\cot^2(\theta)}{r})e_r$ +
$(-0 +\frac{\cos(\phi)\cos(\theta)}{r\sin(\theta} - \frac{\sin(\phi)\sin(\theta)}{r})e_\theta$ +
$(0+ \frac{\sin(\phi)\sin(\theta)}{r}+\frac{\sin(\phi)\sin(\theta)}{r})e_\phi$
Is this correct? Am I missing some identities that would express this in a much simpler way? I've left in the zero values. And if it is correct, how do I begin to find out whether it's conservative? I'm expecting a question like this in an exam I have soon, so I need a pretty quick method of being able to calculate curl and whether it's conservative.
Any help would be appreciated.