Divergence of Cross Product Relation

[advphys]

Homework Statement

The problem is given in the following photo:

Actually I did the first proof but I couldn't get the second relation. (Divergence of E cross H).

Homework Equations

They are all given in the photo. (a) (b) and (c).

The Attempt at a Solution

What I tried is to interchange divergence and cross products as it was given in (a). But I couldn't figure out how I am going to get 2 terms at the end. I also tried to apply the relation in (c), but it does not have any cross product, and I think there is no way to use equation in (b). So how can I prove the equation given at the end by using (a) (b) or (c) without decomposing into components or using Einsteins notation.

 

[Fredrik]

The product rule, as it appears in (c), is a vector equation. Its ith component is $\partial_i (fg)=(\partial_i f)g+f\partial_ig$. If you use the definition of the cross product to rewrite the cross products in the problem, you will encounter expressions of the form $\partial_i (fg)$.

Edit: In this problem, you don't even have to use the definition, since (c) also tells you that if f and g are vector-valued functions, you're allowed to use that $\partial_i (f\cdot g)=(\partial_i f)\cdot g+f\cdot\partial_ig$ and $\partial_i (f\times g)=(\partial_i f)\times g+f\times\partial_i g$.

 

[advphys]

That is right. I didn't think using that for cross product. After that I can use (a) to prove the given relation.

It seems this was a little bit dummy question.

Thank you very much!

 

[Fredrik]

Looking at the problem again, I see that the final sentence tells you NOT to use the definition of the cross product to rewrite it in terms of components. But you can still use the comment I added when I edited my previous post.