Vector field identity derivation using Einstein summation and kronecker delta.

In summary, the conversation discusses a problem from Schwinger's "Classical Electrodynamics" involving vector fields \vec{A}(\vec{r}) and \vec{B}(\vec{r}). The goal is to show that \vec{\nabla}\bullet(\vec{A}\vec{B})=(\vec{A}\bullet\vec{\nabla})\vec{B}+\vec{B}(\vec{\nabla}\bullet\vec{A}). The individual attempts at a solution involve using the unit dyadic and the tensor product between A and B. However, there is confusion over the missing terms in the divergence. Further research is needed for a complete solution."
  • #1
jmwilli25
5
0

Homework Statement



Let [tex]\vec{A}(\vec{r})[/tex]and [tex]\vec{B}(\vec{r})[/tex] be vector fields. Show that

Homework Equations



[tex]\vec{\nabla}\bullet(\vec{A}\vec{B})=(\vec{A}\bullet\vec{\nabla})\vec{B}+\vec{B}(\vec{\nabla}\bullet\vec{A})[/tex]
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.
[tex](A_{j}\partial_{j})_{i}B_{i}+B_{i}(\partial_{j}A_{j})_{i}[/tex]
and I don't know where to go from there. I have scoured Google and Wolfram but I was unable to find any help.
 
Last edited:
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  • #2
I figured it out! You have to use what is called the dyadic.
The unit dyadic is 1=ii+jj+kk
 
  • #3
Well, the tensor product between A and B is

[tex] \vec{A}\otimes\vec{B} = A_i B_j \, e_i \otimes e_j [/tex]

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

What is a vector field identity?

A vector field identity is a mathematical equation that describes the relationship between two vector fields. It is used to simplify calculations and solve problems involving vector fields.

What is Einstein summation?

Einstein summation is a notation used to express mathematical equations involving multiple indices. It uses the summation symbol (∑) to represent the summation of terms over a repeated index, making complex equations easier to write and understand.

What is the Kronecker delta?

The Kronecker delta is a mathematical symbol (δ) used to represent the identity matrix. In vector field identity derivation, it is used to simplify equations involving Kronecker delta functions, which are defined as 1 when the indices are equal and 0 when they are not.

How is the vector field identity derived using Einstein summation and Kronecker delta?

The vector field identity is derived by using Einstein summation notation to express the equations in terms of repeated indices and then using Kronecker delta functions to simplify the equations. This results in a simplified and more concise representation of the original equations.

What are the applications of vector field identity derivation using Einstein summation and Kronecker delta?

This technique is commonly used in fields such as physics, engineering, and mathematics to solve problems involving vector fields. It allows for more efficient and accurate calculations, making it a valuable tool in various scientific and technical applications.

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