I was supposed to figure out what $grad(div(e_r))$ is, where $e_r$ is the unit vector in spherical coordinates. In the following I assume $g:=diag(1,r^2,r^2sin^2(\theta))$ be the metric tensor.
My idea was the following: $e_r=(1,0,0)$ in spherical coordinates. And $div(f)=\frac{1}{\sqrt{det(g)}} \sum_{i=1}^3 \partial_i (\sqrt{det(g)} f^{(i)})$, so I did $div(e_r)=\frac{2}{r}$.
Further we have: $grad(\phi)=\sum_{i=1}^3 e_i \frac{1}{\sqrt{g^{i,i}}}\partial_i \phi$ And arrived for $grad(div(e_r))=\frac{-2}{r^2}e_r$
Furthermore I needed to calculate $rot(e_{\theta})$.I took:$rot(F)=\sum_{i,j,k=1}^3 \epsilon_{i,j,k} \frac{1}{\sqrt{g^{ii}}\sqrt{g^{kk}}}( \partial_j(\sqrt{g^{kk}}F^{k}))e_i$ and arrived at $rot(e_{\theta})=\frac{1}{r}e_{\theta}$. My problem is that I don't know whether my basic equations and starting points are correct? Is there anybody here who could tell me if my ideas were correct?
The metric tensor should have $r^2\sin ^2\theta$ instead of $r^2\sin \theta$.
Your computations for the divergence and the gradient of divergence are correct nonetheless. The last one does not look right. I think you need $g^{jj}$ instead of $g^{ii}$ there; I get $ \pm \frac{1}{r}e_{\phi}$ depending on how you order the coordinates. Compare with (2.80) here.