1. Straightforward using index notation.

2.a) Not conservative

2.b) Conservative away from origin since the curl is zero. However, there is no well-defined potential that is valid for the region of the plane minus the origin.

This can either be seen by the fact that necessarily $\Phi = \tan^{-1}(x/y) + \text{const.}$ or $\Phi = -\tan^{-1}(y/x) + \text{const.}$ from direct integration. Taking the first $\Phi$, we see it is not continuous along $x = 0$. The second $\Phi$ is not continuous along $y = 0$. But any region not including the origin would be fine.

Alternatively, you can see that integration of $\int_C \mb{F} \cdot \de{\mb{r}} \neq 0$ for a closer contour encircling the origin.

2.c) To develop the potential, $\nabla \Phi = \mathbf{F}$, use the gradient in spherical coordinates and assume $\Phi$ only depends on $r$. Show that $\Phi' = rf(r)$ and thus $\nabla \cdot \mb{F} = \Phi'' + 2 \Phi'/r$. The divergence of $\mathbf{F}$ can be most easily computed in Cartesian framework.

2.d) $\Phi = (x^2 + y^2)^2/2$; Newton's equations are $\ddot{x} = 2r^2 x$, $\ddot{y} = 2r^2 y$, $\ddot{z} = 0$

3. See techniques from Fourier Series and PDEs. Let $u$ be the difference of two solutions. Write down the governing equations. Work with the energy formulation of \[ \int_{\partial V} u \nabla u \cdot \de{\mb{S}} = \int_V (|\nabla u|^2 + u \nabla^2 u)\, \de{V}. \]

where $V$ is a sphere. See the last problem of MVC PS8 for a similar treatment.Note that on the boundary of a sphere, $\pd{u}{n} = \pd{u}{r}$.

4. See analogous proof for gravitational case in lectures notes. When the origin is located within $S$, you will need to apply the divergence theorem to a volume that does not include the origin (why?). When the origin is not included within $S$, the divergence theorem can be applied as usual.