Let $f(r)$ be a smooth scalar-valued function of $r=|\mathbf{r}|$, and let $a\in\Bbb{R}^{3}$ be a constant vector. I am trying to use suffix/index notation to calculate the expressions $\nabla\times(\mathbf{r}\times\mathbf{a}f(r))$ and $\nabla\cdot(\mathbf{a}f(r))$. This is my progress on the first expression by considering the $i$th component so far:
\begin{align} [\nabla\times(\mathbf{r}\times\mathbf{a}f(r))]_{i} & = \epsilon_{ijk}\partial x_j[\mathbf{r}\times\mathbf{a}f(r)]_{k} \\[2ex] & = \epsilon_{ijk}\partial x_j\epsilon_{klm}r_{l}a_{m}f(r) \\[2ex] & = \epsilon_{ijk}\epsilon_{klm}((\dfrac{\partial r_{i}}{\partial x_{j}}a_{m}f(r)+x_{l}\dfrac{\partial}{\partial x_{j}}(a_{m}f(r))) \\ \end{align}
After that, I am not sure on how to further simplify it but I am aware that I will be using $\epsilon_{ijk}\epsilon_{klm}=\epsilon_{kij}\epsilon_{klm}=\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl}$ at one point. Appreciate any help I can get, thanks.