1. Mar 11, 2014

### rsammas

1. The problem statement, all variables and given/known data

Show that:

∇x(∇xB) = (B∇)B - ∇ (1/2B2)

2. Relevant equations

r = (x,y,z) = xiei

∂xi/∂xj = δij

r2 = xkxk

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

3. The attempt at a solution

On the LHS I simplified to get:

εijk2/∂xj∂xk

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(∂Bi/∂i)-B
= B(∂Bi/∂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.

2. Mar 12, 2014

### 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!

3. Mar 12, 2014

### rsammas

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

4. Mar 12, 2014

### maajdl

(∇xB)xB = (B∇)B - ∇ (B²/2)

is famous in MHD

5. Mar 12, 2014

### rsammas

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

6. Mar 13, 2014

### maajdl

Showing a proof of what?
The original identity is obviously wrong (∇x(∇xB) = (B∇)B - ∇ (1/2B2) is wrong).

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.

"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?

Last edited: Mar 13, 2014