Answer
a) $\nabla \cdot r=3$
b) $\nabla \cdot (rr)=4r$
c) $\nabla^2 r^3=12r$
Work Step by Step
a) Here, we have $\nabla \cdot r=(\dfrac{\partial }{\partial x}i+\dfrac{\partial }{\partial y}j+\dfrac{\partial}{\partial z}k) \cdot (x i+yj+zk)=1+1+1$
and $\nabla \cdot r=3$
b) $\nabla \cdot (rr)=r \nabla \cdot r+r \cdot \nabla r$
$=(x i+yj+zk) \cdot (\dfrac{x}{r}i+\dfrac{y}{r}j+\dfrac{z}{r}k)$
This gives: $ \dfrac{x^2+y^2+z^2}{r}=r$
Thus, we have: $\nabla \cdot (rr)=3(r)+r=4r$
c) $\nabla r^3=\dfrac{\partial (r^3)}{\partial x}i+\dfrac{\partial(r^3) }{\partial y}j+\dfrac{\partial (r^3)}{\partial z}k$
or, $\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) $
This gives: $\nabla^2 r^3=3 \nabla \cdot (rr)$
Hence, $\nabla^2 r^3=3(4r)=12r$
