What are some useful vector calculus identities for vector fields A and B?

[hhhmortal]

Homework Statement

For arbitrary vector fields A and B show that:


∇.(A ∧ B) = B.(∇∧A) - A.(∇∧B)

The Attempt at a Solution

I considered only the 'i'-axis, by saying that it is perpendicular with A and B and then I expanded both the left and right side out. The working is too much to post here..I didn't manage to prove it. I was hoping someone would know if there's a useful webpage where I can find out more about vector calculus identities.

Thanks!

 

[phsopher]

You can use the fact that

[\mathbf A \times \mathbf B]_i = \epsilon_{ijk} A_j B_k

Where there is a summation over repeated indices and \epsilon_{ijk} is completely antisymmetric under an exchange of indices (Levi-Civita symol). That way it's not difficult to prove. Don't know about web sites though.

 

[hhhmortal]

Yea, although I'm being asked to prove it without using suffix notation..which is much more tedious

 

[phsopher]

It is, but it's still a straightforward calculation, just remember the Leibniz rule and keep track of the indices.

 

[hhhmortal]

Ok I completed the identity without using suffix notation, but now I will to use it:

My working out is the following:

I take the ith component of both sides, I first start with the left side:\epsilonijk δ/δx_{}i(A_j B_k)And then for the right side:

(B_i ε_ijk δA_k/δx_{}j) - (A_i ε_ijk δB_k/δx_{}j)

As you can see the differentials are different, for the left side it is w.r.t the 'i' component while for the right side it is w.r.t 'j' component

 

[phsopher]

There are no components on either side, the expression is a scalar since it's a dot product. There is a summation over i,j and k so the left hand side is

\nabla \cdot (\mathbf A \times \mathbf B) = \sum_{i=1}^3 \partial_i (\mathbf A \times \mathbf B)_i = \sum_{i=1}^3 \partial_i (\sum_{j,k=1}^3\epsilon_{ijk} A_j B_k) = \sum_{i,j,k=1}^3 \epsilon_{ijk} \partial_i (A_j B_k)

Calculate the derivative and rearrange the terms to get the expression on the right hand side (Hint: use the fact that epsilon is antisymmetric under an exchange of indices).

 

[hhhmortal]

There are no components on either side, the expression is a scalar since it's a dot product. There is a summation over i,j and k so the left hand side is

\nabla \cdot (\mathbf A \times \mathbf B) = \sum_{i=1}^3 \partial_i (\mathbf A \times \mathbf B)_i = \sum_{i=1}^3 \partial_i (\sum_{j,k=1}^3\epsilon_{ijk} A_j B_k) = \sum_{i,j,k=1}^3 \epsilon_{ijk} \partial_i (A_j B_k)

Calculate the derivative and rearrange the terms to get the expression on the right hand side (Hint: use the fact that epsilon is antisymmetric under an exchange of indices).

Oh ok! why is k=1 though? I am a bit confused. So you have summed the left side over i, will this mean :

(A_j )(B_z) will be differentiated w.r.t to dx_i and then I can use the product rule.

 

[phsopher]

No, the summation is over all three indices, it has to be because the end result is a scalar so there can't be any indices remaining.What the above means is that i goes from 1 to 3, j goes from 1 to 3 and k goes from 1 to 3.

In any case, yes, differentiate w.r.t dx_i inside the sum and you will get two terms. Rearrange them using the antisymmetricity of epsilon and you will get the two terms of the right hand side.