The Student Room Group
Reply 1
u\mathbf{u} \cdot \nabla is the "scalar" operator u1x1+u2x2+u3x3\displaystyle u_1 \frac{\partial}{\partial x_1} + u_2 \frac{\partial}{\partial x_2} + u_3 \frac{\partial}{\partial x_3}, where uiu_i are the components of u and xix_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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