Index Notation Identity for Vector Fields

[squire636]

Homework Statement

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!

Homework Equations

The Attempt at a Solution

f(x) = ∂_iε_ijkA_jε_kab∂_aB_b - ε_ijk∂_jA_kε_iab∂_aB_b + A_i∂_i∂_jB_j

f(x) = ε_kijε_kab∂_iA_j∂_aB_b - ε_ijkε_iab∂_jA_k∂_aB_b + A_i∂_i∂_jB_j

(note that I changed ε_ijk to ε_kij in the first term)

f(x) = (δ_iaδ_jb - δ_ibδ_ja)∂_iA_j∂_aB_b - (δ_jaδ_kb - δ_jbδ_ka)∂_jA_k∂_aB_b + A_i∂_i∂_jB_j

f(x) = ∂_iA_j∂_iB_j - ∂_iA_j∂_jB_i - ∂_jA_k∂_jB_k + ∂_jA_k∂_kB_j + A_i∂_i∂_jB_j

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.

 

[haruspex]

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

 

[squire636]

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

 

[haruspex]

OK, so you're familiar with the triple product identities? (http://en.wikipedia.org/wiki/Triple_product). Try applying them to either of the first two terms to see if you can make them the same.