I have been trying to prove that $\nabla ^2 r^n = n(n+1)r^{n-2}$ but I've been unsuccessful. I have been using the following identities: $\nabla f(r) = f'(r) \hat {r}$ and $\nabla \cdot (\varphi \vec{F} )= \nabla \varphi \cdot \vec F + \varphi (\nabla \cdot \vec F)$.
Here's my attempt:
$\begin{align} \nabla ^2 r^n &= \nabla \cdot (\nabla r^n ) \\ &=\nabla \cdot (nr^{n-1} \hat{r} ) \\ &= n [ \nabla (r^{n-1}) \cdot \hat {r} + (\nabla \cdot \hat{r}) r^{n-1} ] \\ &= n [(n-1) r^{n-2} + 3 r^{n-2}] \\ &= n(n+2) r^{n-2}. \end{align}$
Can anyone point me where I am going wrong?
$\nabla . \hat {r} = 3/r $ is where you went wrong. Convert it to $\vec {r}/r $ and calculate to see the difference. The rest of the answer is fine.