Give a function $f(r)$ with input $r = (x,y,z)^{\text{T}}$ and a vector field $V = f\cdot r$
How to determine $f$ so that $\nabla \cdot A = 0 \quad \textsf{(condition for solenoidality)}$
I have only reached as far:
$$\nabla \cdot (f\cdot r) = f\cdot \nabla \cdot r + r \cdot \nabla f = 0$$
which yields a complicated partial differential equation:
$$0 = 3\,f + x\cdot \partial_x\,f + y\cdot \partial_y\,f + z\cdot \partial_z\,f $$
Is there any symmetry to exploit?