For the first part, to find $\alpha$ and $ \beta$, I said that the vector field has to be divergion-less so $\text{div}(\mathbf{F})=0$. So I got $\alpha=-\frac{1}{2},\beta=-3$. However, for the second part, I am not sure how to apply Stokes' theorem, how do I find a vector field $\mathbf{G}$ such that $\mathbf{F}=\nabla \times \mathbf{G}$ to reduce the integral to a line integral??
Any help would be appreciated, thanks.