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When I calculus the term $u\cdot \nabla u$, where $u$ is a vector, I get two different results by calculus in different ways i.e.

$((u\cdot \nabla) u)_i=\sum\limits_{k=1}^n(u_k\nabla_k)u_i$ and $(u\cdot (\nabla u))_i=\sum\limits_{k=1}^nu_k(\nabla u)_{ki}=\sum\limits_{k=1}^nu_k(\nabla_iu_k)$.

Which one is right?

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That's for you to decide in defining the notation you use, or to clarify by using parentheses.

I personally would tend to take $u\cdot\nabla u$ to mean $(u\cdot\nabla)u$, because I studied physics and that construction is more common there than the other one. Also, a dot is much more usual for the scalar product of two vectors (the first interpretation) than for the product of a vector and a matrix (the second interpretation). But still it's somewhat ambiguous, and if you can't expect your readers to know which one you mean, parentheses seem advisable.

Anyway, $\nabla u$ is a bad (though common) notation for the gradient of a vector field, since it doesn't fit with the usual rules for matrices and vectors. The “right” way to write it would be $(\nabla u^\top)^\top$, and then there would be no confusion, since $(u\cdot\nabla)u\neq u\cdot(\nabla u^\top)^\top$ and the only consistent way to interptet $u\cdot\nabla u$ would be $(u\cdot\nabla)u$.

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