In my fluidum mechanics course we encounter a lot of vector calculus problems, one of which I am struggling with for a while now. We must prove that:
$$ \vec{V} \cdot (\vec{\nabla}\vec{V}) = (\vec{\nabla}\cdot\vec{V})\vec{V} $$
**solely** using sommation/index notation. $\vec{\nabla}\vec{V}$ is a second order tensor which we denote by: 
$$\left(\sum\limits_{i}\hat{e_i}\frac{\partial}{\partial x_i}\right)\left(\sum\limits_{j}\hat{e_j}v_j\right) $$
I think my confusion lies in the use of $\frac{\partial}{\partial x_i}$ in a tensor since we haven't used tensors commonly before taking this course. Could someone maybe prove this and clarify how second order tensors work in general?