Is $Cr^a\hat r$ always a conservative vector field? Is the vector field $\vec r(r, \theta, \phi) = Cr^a\hat r$, where $C, a\in \Bbb R$ are constants and $r \in \Bbb R^+ \cup \{0\}$ is the radial component, always a conservative vector field?  I really just want to know if I have a force field of this type can I always find a potential function?
Yes. The curl in spherical polars involves $\theta$ and $\phi$ derivatives of the radial component, but not $r$ derivatives. 



Hence $\nabla \times \vec r = 0$ and therefore the vector field is conservative.

Added: As the grad of a scalar field in polar coordinates is given by



...it is easy to integrate your vector field to find a corresponding potential scalar field.
Yes. The curl in spherical polars involves $\theta$ and $\phi$ derivatives of the radial component, but not $r$ derivatives. 



Hence $\nabla \times \vec r = 0$ and therefore the vector field is conservative.

Added: As the grad of a scalar field in polar coordinates is given by



...it is easy to integrate your vector field to find a corresponding potential scalar field.