I realize this may be an elementary question however I want to make sure I am not mistaken in my analysis.
I am uncertain what property of $\nabla$ means these two terms are not equal. (It came up in a fluid mechanics proof related to vorticity). (note: $u, v$ are vectors).
From continuity: $$ \nabla \cdot u = 0$$
Which means: $$ w( \nabla \cdot u)=0$$
However: $$ (u \cdot \nabla)w \neq 0 $$
Is it because the divergence will always act on the RH element first?
Any help is much appreciated