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    What does (\mathbf{u} . \nabla) \mathbf{v} mean? I know that (\mathbf{u} . \nabla) is an operator, but how does it act on \mathbf{v} when there's no cross or dot between the two?

    \mathbf{u} \cdot \nabla is the "scalar" operator \displaystyle u_1 \frac{\partial}{\partial x_1} + u_2 \frac{\partial}{\partial x_2} + u_3 \frac{\partial}{\partial x_3}, where u_i are the components of u and x_i are the coordinates. As such, it acts on v directly without the need for any dot or cross.

    (In any case, "vector" operators can act directly on vectors as well; the result is something which might be regarded as a matrix or a tensor of rank 2.)

    The operator acts on each component of v.
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