Compute $(\mathbf{A} \cdot \mathbf{\nabla})\mathbf{B}$

Question

I'm currently reading Intro to Electrodynamics by Griffiths, and in the maths section, there is the following problem:

"If $\mathbf{A}$ and $\mathbf{B}$ are two vector functions, what does the expression $(\mathbf{A} \cdot \mathbf{\nabla})\mathbf{B}$ mean?

(That is, what are its $x$, $y$, and $z$ components, in terms of the Cartesian components of $\mathbf{A}$, $\mathbf{B}$, and $\mathbf{\nabla}$?"

First, I naively thought it was just the divergence of $\mathbf{A}$ multiplied by $\mathbf{B}$, and that for some reason Griffiths wrote the divergence as $\mathbf{A} \cdot \mathbf{\nabla}$ this time, rather than $\mathbf{\nabla} \cdot \mathbf{B}$, which is the way i'm used to seeing it.

But when i looked in the solution manual, it says the answer is

$
\begin{align}
(\mathbf{A} \cdot \mathbf{\nabla}) &= \left(
A_x \frac{\partial B_x}{\partial x} + A_y \frac{\partial B_x}{\partial y} + A_z \frac{\partial B_x}{\partial z}
\right) \mathbf{\hat{x}} \\ &+
\left(
A_x \frac{\partial B_y}{\partial x} + A_y \frac{\partial B_y}{\partial y} + A_z \frac{\partial B_y}{\partial z}
\right) \mathbf{\hat{y}} \\ &+
\left(
A_x \frac{\partial B_z}{\partial x} + A_y \frac{\partial B_z}{\partial y} + A_z \frac{\partial B_z}{\partial z}
\right) \mathbf{\hat{z}}
\end{align}
$

I thought this question was weird because Griffiths hadn't used this notation yet until now, so I'm not sure why he thought I would be able to do this problem. I know it's not the typo or anything, because the next problem is similar, as it wants me to find $(\mathbf{\hat{r}} \cdot \mathbf{\nabla})\mathbf{\hat{r}}$

So, I guess my question is, what does this expression mean, and how do I calculate it? Obviously I'm not going to memorise this mess, and the notation seems to suggest a dot product is somehow involved.

Answer 1

It means that the differential operator $$ \mathbf{A} \cdot \nabla = (A_x,A_y,A_z) \cdot (\partial_x,\partial_y,\partial_z) = A_x \partial_x + A_y \partial_y + A_z \partial_z $$ acts componentwise on the vector $\mathbf{B}$.