By chance, I have seen this post which has something different from my notes (though I have used different notations).
$$\nabla(f\boldsymbol{v}) =\partial_i(fv_j)=(\partial_if)v_j+f\partial_iv_j=(\nabla f)\otimes\boldsymbol{v}+f\nabla\boldsymbol{v}$$ But in the post, it is transposed $\nabla(a\boldsymbol{v}) = \boldsymbol{v}\otimes\nabla a + a\nabla\boldsymbol{v}$ ! When I check again, I found the confusion since comes from $\nabla(f\boldsymbol{v})\equiv\nabla(\boldsymbol{v}f)$, but how it is possible to have two answers which are different?
