- #1

DougD720

- 47

- 0

## Homework Statement

Show: [itex]∇(\vec{A} . \vec{B}) = \vec{B} \times (∇ \times \vec{A}) + (\vec{B} \times ∇)\vec{A} + \vec{A} \times (∇ \times \vec{B}) + (\vec{A} \times ∇)\vec{B}[/itex]

## Homework Equations

I tried tackling this sucker using cartesian coordinates but that's not the way to go. I believe the way to go is to use Levi-Civita Tensors and Delta Functions (he introduced us to them on the first day of class but I don't understand how to manipulate them):

[itex]\vec{A} . \vec{B} = A_{i}B_{j}δ_{ij}[/itex]

[itex]\vec{A} \times \vec{B} = ε_{ijk}A_{j}B_{k}[/itex]

## The Attempt at a Solution

So I've been at this thing for hours (and this is the second time I've tried posting it my computer got funny after the first time - this is going to be fun to Latex in again :( )

Here's what I've got:

[itex]LHS = ∇(A_{i}B{j}δ_{ij})[/itex]

[itex]RHS = (B_{i}A_{j}δ_{ij})∇ - (B_{i}∂_{j}δ_{ij})\vec{A} + (ε_{ijk}B_{j}∂_{k})\vec{A} + (A_{i}B_{j}δ_{ij})∇ - (A_{i}∂_{j}δ_{ij})\vec{B} + (ε_{ijk}A_{j}∂_{k})\vec{B}[/itex]

=> [itex]RHS = 2∇(A_{i}B_{j}δ_{ij}) = (ε_{ijk}B_{j}∂_{k} - B_{i}∂_{j}δ_{ij})\vec{A} + (ε_{ijk}A_{j}∂_{k} - A_{i}∂_{j}δ_{ij})\vec{B}[/itex]

This is the first part of the problem, the second asks to show another identity but if someone could show me / I could figure out how to solve this one I'd definitely be able to solve the second part. I just don't really get how to work with these Delta Functions and Levi-Civita Tensors.

Thanks!