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How to integrate this by parts?

  1. May 7, 2015 #1
    • THIS HOMEWORK PROBLEM WAS SUBMITTED TO THE WRONG FORUM, AND THERE IS NO TEMPLATE.
    Hey guys,

    So here's the issue I'm faced with. I need to integrate the following by parts (twice):

    [itex]\int d^{3}y\, e^{ik(\hat{n}_{0}-\hat{n})\cdot\vec{y}}\left[ \nabla\times\nabla\times\hat{\epsilon}_{0} \right][/itex]

    And I have absolutely no clue how to approach this. The result I'm meant to reach is proportional to

    [itex]\int d^{3}y\, e^{ik(\hat{n}_{0}-\hat{n})\cdot\vec{y}} \hat{n}\times\hat{n}\times\hat{\epsilon}_{0}[/itex]

    The hats denote unit vectors I believe.

    I don't know how to integrate by parts an expression involving the curl operator...can someone help please?

    Thanks!
     
  2. jcsd
  3. May 7, 2015 #2

    fzero

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    Integration by parts in the usual case works because of the product rule for differentiation:

    $$ \frac{d}{dx} (f g) = \frac{df}{dx} g + f \frac{dg}{dx} .$$

    In your case, you need to use some definition of the curl operation to show that

    $$ \nabla \times (f \vec{A}) = \nabla f \times \vec{A} + f (\nabla \times \vec{A}).$$

    You can then manipulate this in a way analogous to the usual integration by parts formula to find an appropriate generalization.
     
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