How to rewrite $\frac{\partial \rho u_i u_j}{\partial x_j}$ in vector notation I want to rewrite this index notation expression to a vector notation /symbolic notation. 
$$\frac{\partial \rho u_i u_j}{\partial x_j}=\frac{\partial \rho }{\partial x_j}u_i u_j+\rho\frac{\partial u_i}{\partial x_j} u_j+\rho u_i\frac{\partial u_j}{\partial x_j} $$
This is what i have tried so far:
Expression 1:
$$=u_i\vec{u}\cdot\left(\nabla\rho\right)+\rho\vec{u}\cdot\left(\nabla\vec{u}\right)+\rho u_i\left(\nabla \cdot \vec{u}\right)$$
$$=u_i\left[\nabla \cdot\left(\rho\vec{u}\right)\right]+\rho\vec{u}\cdot\left(\nabla\vec{u}\right)$$
Expression 2:
$$=\nabla\cdot\left(\rho u_i \vec{u}\right)$$

Given that I prefer index notation, that presents less ambiguities, I would write expression 1 as:
$$
\mathbf{u}(\mathbf{u}\cdot\nabla)\rho+\rho(\mathbf{u}\cdot\nabla)\mathbf{u}+\rho\mathbf{u}(\nabla\cdot\mathbf{u})
$$
and expression 2 as
$$
\nabla\cdot(\rho\mathbf{u}\otimes\mathbf{u})
$$
