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**Ackbeet** 2. The expression $\displaystyle \mathbf{u}\cdot\Delta$ is an operator, and with operators, order matters. That is,

$\displaystyle \mathbf{u}\cdot\Delta\not=\dfrac{\partial u_{x}}{\partial x}+\dfrac{\partial u_{y}}{\partial y}+\dfrac{\partial u_{z}}{\partial z},$ but

$\displaystyle \mathbf{u}\cdot\Delta=u_{x}\,\dfrac{\partial }{\partial x}+u_{y}\,\dfrac{\partial }{\partial y}+u_{z}\,\dfrac{\partial }{\partial z}.$

You then have

$\displaystyle (\mathbf{u}\cdot\Delta)\mathbf{u}=\left(u_{x}\,\df rac{\partial }{\partial x}+u_{y}\,\dfrac{\partial }{\partial y}+u_{z}\,\dfrac{\partial }{\partial z}\right)\mathbf{u}.$

Make sense?