How to Write ∇ × (∇ × A) in Einstein Notation?

[smoking-frog]

Homework Statement

Write $$\nabla \times (\nabla \times \vec A)$$ in Einstein-Notation, whereas $$\vec A$$ is the vector potential of the magnetic field.

Homework Equations

$$(\vec a \times \vec b)=\varepsilon_{ijk} a_j b_k$$

The Attempt at a Solution

$$\nabla \times (\nabla \times \vec A)=\varepsilon_{ijk} \partial_j(\varepsilon_{lmn}\partial_m A_n)_k$$

What to do with $$(\varepsilon_{lmn}\partial_m A_n)_k$$ though?

Thank you for your help!

 

[CompuChip]

Homework Equations

$$(\vec a \times \vec b)=\varepsilon_{ijk} a_j b_k$$

On the right hand side, there is summation over j and k, but i is a free index which only occurs once. What you really meant to write, then, is

$$(\vec a \times \vec b)_i=\varepsilon_{ijk} a_j b_k$$

 

[andrien]

In index notation it reads
ε_ijk∂_j(ε_klm∂_lu_m)
after it you can use an identity
ε_kijε_klm=δ_ilδ_jm-δ_imδ_jl
(I hope this is not complete solution.if it is then delete this)