I'm trying to understand vector and tensor notation, but I'm coming across some difficulties. Say I have vector $\vec{u}$ and I compute its gradient $\nabla \vec{u}$. Then I get a tensor $\frac{\partial u_i}{\partial x_j}$. If I write explicitly all its components, I get $$\nabla {\bf u} = \begin{Bmatrix} \frac{\partial u_1}{\partial x_1} & \frac{\partial u_1}{\partial x_2} & \frac{\partial u_1}{\partial x_3}\\ \frac{\partial u_2}{\partial x_1} & \frac{\partial u_2}{\partial x_2} & \frac{\partial u_2}{\partial x_3}\\ \frac{\partial u_3}{\partial x_1} & \frac{\partial u_3}{\partial x_2} & \frac{\partial u_3}{\partial x_3}\\ \end{Bmatrix}.$$ I'm aware that generally tensors are not the same thing as a matrix. How shall I deal with this quantity though: is it a matrix?
If I compute $\vec{u} \cdot \nabla \vec{u} = u_j \frac{\partial u_i}{\partial x_j}=s_i$, where $s_i$ the different component of a vector ($s_1$, $s_2$, $s_3$)? However, if I were to do this as matrix manipulation (row by columns) $$(u_1,u_2, u_3) \cdot \begin{Bmatrix} \frac{\partial u_1}{\partial x_1} & \frac{\partial u_1}{\partial x_2} & \frac{\partial u_1}{\partial x_3}\\ \frac{\partial u_2}{\partial x_1} & \frac{\partial u_2}{\partial x_2} & \frac{\partial u_2}{\partial x_3}\\ \frac{\partial u_3}{\partial x_1} & \frac{\partial u_3}{\partial x_2} & \frac{\partial u_3}{\partial x_3}\\ \end{Bmatrix},$$ I would get $s_i= u_i \frac{\partial u_i}{\partial x_j}.$ Could someone help me to understand to write component and matrix forms in consistent and correct form?
