What is the difference between $ (A\cdot\nabla )B$ and $A(\nabla\cdot B)$, where $A$ and $B$ are two vectors.
My guess is that $ (A\cdot\nabla )B$ means the vector $A$ is being multiplied by the rate of change of $B$ in the direction of $A$. And $A(\nabla\cdot B)$ means vector $A$ is being multiplied by the divergence of $B$.