Vector Cross Product With Its Curl

[John Delaney]

Homework Statement

Prove A X (∇ X A) = ∇(A²/2) - A · ∇A

Relevant Equations

εijk εlmk = δil δjm - δim δjl

Starting with LHS:

ê_i ε_ijk A_j (∇xA)_k

ê_i ε_ijk ε_lmk A_j (d/dx_l) A_m

(δ_il δ_jm - δ_im δ_jl) A_j (d/dx_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_j (d/dx_i) A_j ê_i - A_j (d/dx_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):

∇(A²) - A · ∇A

 

[mfb]

What is the derivative of f(x)=x²/2?
Your expression here is analogous.

 

[rmwhitehill7918]

Notice that $A^2 = A \cdot A$. When the gradient operator is applied to this term, you get two terms looking like this: $A(\nabla A)$. In your initial work, you only have one of these terms, so to account for the duplicate, you must divide $\nabla A^2$ by two, which results in the term $\nabla \frac{A^2}{2}$ appearing in the identity you are attempting to prove.