Answer
a) $\nabla \cdot r=3$
b) $\nabla \cdot (rr)=4r$
c) $\nabla^2 r^3=12r$
Work Step by Step
(a) $\nabla \cdot r=(\dfrac{\partial i }{\partial x}+\dfrac{\partial j }{\partial y}+\dfrac{\partial k}{\partial z}) \cdot (x i+yj+zk)=1+1+1=3$
(b) $\nabla \cdot (rr)=r \nabla \cdot r+r \cdot \nabla r=(x i+yj+zk) \cdot (\dfrac{x}{r}i+\dfrac{y}{r}j+\dfrac{z}{r}k) =x^2+y^2+z^2/r=3r+r=4r$
(c) $\nabla r^3=\dfrac{\partial (r^3)}{\partial r} \times (\dfrac{\partial r}{\partial x}i+\dfrac{\partial r}{\partial y}j+\dfrac{\partial r}{\partial z}k)=(3r^2) \nabla r =3(4r)=12r$
