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## Homework Statement:

- Prove A X (∇ X A) = ∇(A²/2) - A · ∇A

## Relevant Equations:

- εijk εlmk = δil δjm - δim δjl

Starting with LHS:

ê

ê

(δ

δ

At this point, the LHS should equal the RHS in the problem statement, but I have no clue where the 1/2 comes from...been trying to figure it out for the past few hours at this point. Rather, I get this and I'm not sure where my lapse in understanding is (I'm relatively new to index notation):

∇(

ê

_{i}ε_{ijk}__A__(∇x_{j}__A__)_{k}ê

_{i}ε_{ijk}ε_{lmk}__A__(d/dx_{j}_{l})__A___{m}(δ

_{il}δ_{jm}- δ_{im}δ_{jl})__A__(d/dx_{j}_{l})__A___{m}ê_{i}δ

_{il}δ_{jm}__Aj__(d/dx_{l})__A___{m}ê_{i}- δ_{im}δ_{jl}__Aj__(d/dx_{l})__A___{m}ê_{i}__A__(d/dx_{j}_{i})__A___{j}ê_{i}-__A__(d/dx_{j}_{j})__A___{i}ê_{i}At this point, the LHS should equal the RHS in the problem statement, but I have no clue where the 1/2 comes from...been trying to figure it out for the past few hours at this point. Rather, I get this and I'm not sure where my lapse in understanding is (I'm relatively new to index notation):

∇(

__A__²) -__A__· ∇__A__