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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

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