What is the meaning of $A. \nabla $?

Question

Suppose you have a vector field $A = A_1 \hat{i} + A_2\hat{j}+ A_3\hat{k} $. Then $\nabla \cdot A $ would represent the divergence. But what does $A \cdot \nabla$ mean below, and what would it come out to be? The relevant problems are (b) and (d) below.

Answer 1

If $\mathbf A = \pmatrix{a_x\\a_y\\a_z}$, then $$(\mathbf A\cdot\nabla)\phi = a_x\frac{\partial}{\partial x}\phi + a_y\frac{\partial}{\partial y}\phi + a_z\frac{\partial}{\partial z}\phi$$

Basically, you treat $\nabla$ as a vector of derivatives and do vector algebra, except that you are careful not to move terms across the derivative.