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Looking at the application of divergence in Cartesian coordinates in Wikipedia I was wondering about the meaning of $\vec A \cdot \nabla$?

This dot product is found at the vector cross product identity: $\nabla \times (\mathbf{A} \times \mathbf{B}) = \mathbf{A} (\nabla \cdot \mathbf{B}) - \mathbf{B} (\nabla \cdot \mathbf{A}) + (\mathbf{B} \cdot \nabla) \mathbf{A} - (\mathbf{A} \cdot \nabla) \mathbf{B}$

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    The 3rd and 4th addends the in the second [identity for vector cross product](http://en.wikipedia.org/wiki/Vector_calculus_identities#Vector_cross_product).2012-03-16

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$\vec A \cdot \nabla = \sum_{i=1}^3 A_i \frac{\partial}{\partial x_i}$

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    So for any vectors $\mathbf{a}$ and $\mathbf{b}$ we have commutativity of the scalar product: $\mathbf{a}\cdot \mathbf{b}=\mathbf{b}\cdot \mathbf{a}$ but not with the del operator since $\mathbf{a}\cdot \boldsymbol{\nabla}\neq \boldsymbol{\nabla}\cdot \mathbf{a}$??2017-10-24