question on vector calculus notation

Question

I just have a question about the vector calculus notation: $$(u \cdot \nabla)u$$ Is that the same as $( \nabla \cdot u)u$?

Answer 1

No, these are not the same. The vector $(u\cdot\nabla)u$ is the directional derivative of $u$ in the direction of $u$. It may not be (and probably isn't) parallel to $u$ at each point. The vector $(\nabla \cdot u)u$ is $u$ multiplied by its divergence. It is always parallel to $u$ at each point.