# Advanced Vecor calculus - Cross Product

1. Jan 12, 2012

### EmmaLemming

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

For arbitrary vector fields A and B show that;

∇ ^ (A^B) = (∇ . B)A - (∇.A)B + (B.∇)A - (A.∇)B

2. Relevant equations

where (A.∇)B = ((A.∇)Bx, (A.∇)By, (A.∇)Bz)

and (A.∇)f = Ax δf/δx + Ay δf/δy + Az δf/δz = (Ax, Ay, Az)( δf/δx, δf/δy, δf/δz)f

3. The attempt at a solution

I asked my lecturer and he said something about finding the x-component and wrote

δ/dy(AyBz - AzBy) - δ/δz(AxBy - AyBx)

on my work but I don't really see how this applies...

Any help would be really appreciated :D

2. Jan 12, 2012

### lanedance

how about wirting it out component by component? Long, but is a useful exercise

Also I am assuming "^" means cross product?

3. Jan 12, 2012

### EmmaLemming

I thought that but I don't know how to write it out like that..

Is my lecturer correct in his definiton of the x-component? and if so do you know how he got it to be that?

and yes ^ means cross product I didn't want any confusion with multiplication.

4. Jan 12, 2012

### lanedance

Last edited: Jan 12, 2012