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Divergence of a cross product

  1. Oct 2, 2015 #1
    1. The problem statement, all variables and given/known data
    The problem is given in the following photo:
    problem.png
    Actually I did the first proof but I couldn't get the second relation. (Divergence of E cross H).

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

    3. 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.
     
  2. jcsd
  3. Oct 2, 2015 #2

    Fredrik

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    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##.
     
  4. Oct 2, 2015 #3
    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!
     
  5. Oct 2, 2015 #4

    Fredrik

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    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.
     
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