Given $$\nabla\cdot(\textbf{r}\times\nabla f)~=~(\nabla\times\textbf{r})\cdot\nabla f~-~\textbf{r}\cdot(\nabla\times\nabla f)$$
I would split the equation into 2 part:
$(1)~~~~~~~~~(\nabla\times\textbf{r})\cdot\nabla f = [\epsilon_{ijk}\frac{\partial x_k}{\partial x_j}]\partial_if=0\cdot\partial_if = 0$
$(2)~~~~~~~~~\textbf{r}\cdot(\nabla\times\nabla f)=x_i~\epsilon_{ijk}~\frac{\partial}{\partial x_j}~\partial_k~f=(-\epsilon_{kji}\delta_{ji})~\partial_k~f = 0 \cdot \partial_k~f =0$
Therefore $\nabla\cdot(\textbf{r}\times\nabla f)~=~0$.
Here are the questions:
(1) Is it alright if I split it into half
(2) I would like to know the comments on using the notation