I am currently learning on tensor calculus and having difficulty in understanding the following derivation.
Suppose that $u = (u_1,u_2,u_3)^T$, one of the formula is $$ \begin{align*} \left(u \cdot \nabla u\right)\cdot u &= \left(e_iu_i\cdot\frac{\partial{u_j}}{\partial{x_k}}e_k\otimes e_j\right)\cdot u_me_m\\ &=\left(u_i\frac{\partial{u_j}}{\partial{x_k}}e_i\cdot e_k\otimes e_j\right)\cdot u_me_m\\ &=\left(u_i\frac{\partial{u_j}}{\partial{x_i}}e_j\right)\cdot u_me_m\\ &=u_iu_m\frac{\partial{u_j}}{\partial{x_i}}e_j\cdot e_m\\ &=u_iu_j\frac{\partial{u_j}}{\partial{x_i}} \end{align*} $$ the other one is $$ \begin{align*} \nabla\cdot u\frac{|u|^2}{2} &= \left(e_i\frac{\partial{}}{\partial{x_i}}\cdot u_je_j\right)\frac{u_ku_k}{2}\\ &= \frac{\partial{u_i}}{\partial{x_i}}\frac{u_ku_k}{2} \end{align*} $$
I just want to know whether my derivation is correct or not cause I was confused that which of the following is correct?
$$ \begin{align} \nabla u &= \frac{\partial{u_i}}{\partial{x_j}}e_j\otimes e_i\\ \nabla u &= \frac{\partial{u_i}}{\partial{x_j}}e_i\otimes e_j\\ \end{align} $$