what does $(A\cdot\nabla)B$ mean?

Question

I was studying a physics book and I saw this expression $$(A\cdot\nabla)B$$ where $A$ and $B$ are vectors. What's the definition of this?

I've also seen this in some identities

Answer 1

The idea is that, say, $A=(A_1,A_2,A_3)$ and $\nabla = (\partial/\partial x_1,\partial/\partial x_2,\partial/\partial x_3)$, so formally $$A\cdot\nabla=A_1\frac{\partial}{\partial x_1}+A_2\frac{\partial}{\partial x_2}+A_3\frac{\partial}{\partial x_3},$$so$$(A\cdot\nabla)B=A_1\frac{\partial B}{\partial x_1}+A_2\frac{\partial B}{\partial x_2}+A_3\frac{\partial B}{\partial x_3}.$$

Answer 2

We can expand the vector notation for $\left(\vec A\cdot \nabla\right)\vec B$ as

$$\left(\vec A\cdot \nabla\right)\vec B=\left(A_x\frac{\partial \vec B}{\partial x}+A_y\frac{\partial \vec B}{\partial y}+A_z\frac{\partial \vec B}{\partial z}\right)$$

In tensor notation, this can be more compactly written as

$$\left(\left(\vec A\cdot \nabla\right)\vec B\right)_i=A_j\partial_j(B_i)$$

where summation over the index $j$ is implied.

Answer 3

If you're in $\Bbb{R}^3$ (usually the case in physics) and $\mathbf{A}$ is a vector and ${B}$ is a scalar

$$\begin{align}\mathbf{A}=&(A_x,A_y,A_z)\\\nabla=&(\nabla_x,\nabla_y,\nabla_z)\\(\mathbf{A}\cdot\nabla)=&A_x{\partial\over \partial{x}}+A_y{\partial\over \partial{y}}+A_z{\partial\over \partial{z}}\end{align}$$

And so

$$(\mathbf{A}\cdot\nabla){B}=A_x{\partial{B}\over \partial{x}}+A_y{\partial{B}\over \partial{y}}+A_z{\partial{B}\over \partial{z}}$$