Solve Grad x (grad x B) Equation

[rsammas]

Homework Statement

Show that:

∇x(∇xB) = (B∇)B - ∇ (1/2B²)

Homework Equations

r = (x,y,z) = x_ie_i

∂x_i/∂x_j = δ_ij

r² = x_kx_k

δ_ij = 1 if i=j, 0 otherwise (kronecker delta)
ε_ijk is the alternating stress tensor and summⁿ convⁿ is assumed.

The Attempt at a Solution

On the LHS I simplified to get:

ε_ijk∂²/∂x_j∂x_k

but was unsure what to do next because the RHS contains only first order derivatives

On the RHS I was able to get to:

(B∇)B - ∇ (1/2B²) = B(∂B_i/∂i)-B
= B(∂B_i/∂i-1)

I feel like I'm just not seeing some simple trick, or there is a rule that I don't remember/haven't learned. This is for my Classical Mechanics class BTW.

 

[vanhees71]

There must be something wrong in your problem statement or how can you get an expression which is quadratic in \vec{B} taking derivatives of an expression that contains only one \vec{B}? The correct equation to prove is
\vec{\nabla} \times (\vec{\nabla} \times \vec{B})=\vec{\nabla} (\vec{\nabla} \cdot \vec{B}) - \vec{\nabla}^2 \vec{B},
which holds, however, only in Cartesian coordinates!

 

[rsammas]

that's what I was thinking, but the assignment is what I wrote above

 

[maajdl]

(∇xB)xB = (B∇)B - ∇ (B²/2)

is famous in MHD

 

[rsammas]

that's still not what I'm asking. but maybe showing a proof might help me out a bit

 

[maajdl]

Showing a proof of what?
What are you asking, actually?
The original identity is obviously wrong (∇x(∇xB) = (B∇)B - ∇ (1/2B2) is wrong).
Shall we advise you to complain to your teacher?

The proof of the second identity, (∇xB)xB = (B∇)B - ∇ (B²/2), is straightforward by using components representation.

Using the notation "eik" for the Levi-Civita tensor,
using 'F,l" to denote the derivative of F with respect to xl,
(∇xB)xB can be developed as follows:

((∇xB)xB)i
= eijk (ejlm Bm,l) Bk
= - eikj elmj Bm,l Bk
= -(eil ekm - eim ekl) Bm,l Bk
= - Bk,i Bk + Bi,k Bk
= - (Bk²/2),i + Bi,k Bk

which ends the proof.

Reading you initial post:

"εijk is the alternating stress tensor ..."
"On the LHS I simplified to get: εijk∂2/∂xj∂xk"

I have the feeling you lack some basic understanding, since it makes almost no meaning.
I don't know if your question is part of a math course or a physics course (electrodynamics).
In any case, you need to go back to the basics.
The strange thing is that the identity "(∇xB)xB = (B∇)B - ∇ (B²/2)" is indeed related to the Maxwell stress tensor in electrodynamics (if B is the magnetic field). The second term is then called the magnetic pressure.
As you posted in the "Calculus & Beyond Homework" section I wonder how you could have mixed that "math exercise" with electrodynamics. Is Google the reason?