Vector field identity derivation using Einstein summation and kronecker delta.

[jmwilli25]

Homework Statement

Let $$\vec{A}(\vec{r})$$and $$\vec{B}(\vec{r})$$ be vector fields. Show that

Homework Equations

$$\vec{\nabla}\bullet(\vec{A}\vec{B})=(\vec{A}\bullet\vec{\nabla})\vec{B}+\vec{B}(\vec{\nabla}\bullet\vec{A})$$
This is EXACTLY how it is written in Ch 3 Problem 2 of Schwinger "Classical Electrodynamics"

The Attempt at a Solution

The only thing I can think of doing is starting from the right hand side and trying to get back to right. This is because I have never seen the left side written like that.
$$(A_{j}\partial_{j})_{i}B_{i}+B_{i}(\partial_{j}A_{j})_{i}$$
and I don't know where to go from there. I have scoured Google and Wolfram but I was unable to find any help.

 

[jmwilli25]

I figured it out! You have to use what is called the dyadic.
The unit dyadic is 1=ii+jj+kk

 

[dextercioby]

Well, the tensor product between A and B is

$$\vec{A}\otimes\vec{B} = A_i B_j \, e_i \otimes e_j$$

The divergence is acting on both A and B, so I don't get why the other 2 terms are missing.