In my fluid mechanics course we encounter a lot of vector calculus problems, one of which I have been struggling with for a while now. We must prove that $$ \vec{V} \cdot \left(\vec{\nabla}\vec{V}\right) = \left(\vec{\nabla}\cdot\vec{V}\right)\vec{V} $$ solely using summation/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?