`\text{rot}(((\mathbf{ar})(\mathbf{a} times \mathbf{r}))/(r^N))=`

`epsilon_{ijk} partial_j (((\mathbf{ar})(\mathbf{a} times \mathbf{r}))/(r^N))_k=`

`epsilon_{ijk} partial_j (a_n x_n (\mathbf{a} times \mathbf{r})_k)/(r^N)=`

`epsilon_{ijk} partial_j a_n x_n epsilon_{klm} a_l x_m r^-N=`

`epsilon_{ijk} epsilon_{klm} a_n a_l partial_j x_n x_m r^-N=`

`epsilon_{ijk} epsilon_{klm} a_n a_l [(partial_j x_n) x_m r^-N+ x_n partial_j(x_m r^-N)]=`

`epsilon_{ijk} epsilon_{klm} a_n a_l [delta_{jn} x_m r^-N+ x_n [(partial_j x_m) r^-N+x_m partial_j r^-N]]=`

`epsilon_{ijk} epsilon_{klm} a_n a_l [delta_{jn} x_m r^-N+ x_n delta_{jm} r^-N+x_n x_m (-N) r^-(N+1)partial_j r]=`

`epsilon_{ijk} epsilon_{klm} a_n a_l [delta_{jn} x_m r^-N+ x_n delta_{jm} r^-N-Nx_n x_m r^-(N+2)x_j]=`

`(delta_{il} delta_{jm} - delta_{jl} delta_{im}) a_n a_l r^-N [delta_{jn} x_m + delta_{jm} x_n -Nx_j x_n x_m r^-2]=`

`(a_i delta_{jm} - a_j delta_{im}) a_n r^-N [delta_{jn} x_m + delta_{jm} x_n -Nx_j x_n x_m r^-2]=`

`a_n r^-N [a_i delta_{jm}delta_{jn} x_m + a_i delta_{jm}delta_{jm} x_n -Na_i delta_{jm}x_j x_n x_m r^-2``-`` a_j delta_{im}delta_{jn} x_m - a_j delta_{im} delta_{jm} x_n +Na_j delta_{im}x_j x_n x_m r^-2]=`

`a_n r^-N [a_i x_n + 3a_i x_n -Na_i x_j x_n x_j r^-2- a_n x_i - a_i x_n +Na_j x_j x_n x_i r^-2]=`

`r^-N [3a_i a_nx_n -Na_i a_nx_n x_j x_j r^-2- a_na_n x_i +Na_j x_j a_nx_n x_i r^-2]=`

`r^-N [3a_i \mathbf{ar} -Na_i \mathbf{ar} \mathbf{r}^2 r^-2- \mathbf{a}^2 x_i +N(\mathbf{ar})^2 x_i r^-2]=`

`r^-N [3a_i \mathbf{ar} -Na_i \mathbf{ar} - \mathbf{a}^2 x_i +N(\mathbf{ae})^2 x_i]=`

`r^-N [(3-N) \mathbf{ara} +(N (\mathbf{ae})^2-\mathbf{a}^2) \mathbf{r}]=`

`r^-N[[(3-N) \mathbf{ar} a_1 +(N (\mathbf{ae})^2-\mathbf{a}^2) x],[(3-N) \mathbf{ar} a_2 +(N (\mathbf{ae})^2-\mathbf{a}^2) y],[(3-N) \mathbf{ar} a_3 +(N (\mathbf{ae})^2-\mathbf{a}^2) z]]`