Curl of (A*B)

1. Nov 24, 2009

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

Prove that $$\nabla\times(A\times B)= (B.\nabla)A-(A.\nabla)B-B(\nabla.A)+A(\nabla.B)$$

2. Relevant equations

bac-cab $$\nabla\times(A\times B)= (\nabla.B)A-(\nabla.A)B$$

3. The attempt at a solution

I know that $$B(\nabla.A)=(\nabla.A)B$$ and $$A(\nabla.B)= (\nabla.B)A$$

So what about $$(B.\nabla)A-(A.\nabla)B=?$$ Does it equal to zero? Or maybe bac cab is not related to this problem!

2. Nov 24, 2009

WiFO215

When and how is this ["bac - cab"] equation valid? The above equation is valid only as long as "a" is....?

3. Nov 24, 2009

4. Nov 24, 2009

LCKurtz

Prove it by letting $\vec A = \langle f,g,h\rangle,\ \vec B = \langle u,v,w\rangle$ and just work out both sides to check they are equal. It's easy, just a little algebra.

5. Nov 24, 2009

WiFO215

Another interesting way would be to do to use a gem of a trick that I learnt off of the Feynman Lectures. Refer to Feynman Lectures, Volume II, Lecture 27, Field Energy and Field Momentum.

6. Nov 24, 2009

Thank you very much for your help. I actually proved it in the way you suggested me, but only for the x-component, and it was a lot of algebra!

7. Nov 24, 2009

Thank you very much. I will try to study it.

8. Nov 25, 2009

Pengwuino

Do you know index notation? Vector identities are quite easy with it.

9. Nov 25, 2009

Please tell me about "index notation". I went to http://en.wikipedia.org/wiki/Index_notation but I didn't completely understand your point of view!

10. Nov 25, 2009

Pengwuino

http://www.physics.ucsb.edu/~physCS33/spring2009/index-notation.pdf [Broken]

Give this a try. Unfortunately, when I learned it, it was during lectures and not in our textbook so I can't tell you what book you can learn it out of. This should be enough though.

Last edited by a moderator: May 4, 2017
11. Nov 26, 2009