# Index Notation Indentity

1. Nov 1, 2012

### squire636

1. The problem statement, all variables and given/known data

Simplify the following, where A and B are arbitrary vector fields:

f(x) = ∇$\bullet$[A $\times$ (∇ $\times$ B)] - (∇ $\times$ A)$\bullet$(∇ $\times$ B) + (A $\bullet$ ∇)(∇ $\bullet$ B)

I know that the correct solution is A $\bullet$ ∇2B, according to my professor. However, I can't get that. I think my mistake is in the first couple of lines, but I'll write out my entire solution and hopefully someone can tell me where I messed up. Thanks!

2. Relevant equations

3. The attempt at a solution

f(x) = ∂iεijkAjεkabaBb - εijkjAkεiabaBb + AiijBj

f(x) = εkijεkabiAjaBb - εijkεiabjAkaBb + AiijBj

(note that I changed εijk to εkij in the first term)

f(x) = (δiaδjb - δibδja)∂iAjaBb - (δjaδkb - δjbδka)∂jAkaBb + AiijBj

f(x) = ∂iAjiBj - ∂iAjjBi - ∂jAkjBk + ∂jAkkBj + AiijBj

Now the first term cancels with the third term, and the second term cancels with the fourth term, so we are left with:

f(x) = (A $\bullet$ ∇)(∇ $\bullet$ B)

But apparently this isn't right.

2. Nov 1, 2012

### haruspex

Are you allowed to use the properties of the triple product: a.(bxc) = b.(cxa) etc?

3. Nov 1, 2012

### squire636

We're allowed to use pretty much whatever we want, as long as I understand it and it makes sense.

4. Nov 1, 2012