How to integrate this by parts?

[Dixanadu]

Hey guys,

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

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

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

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

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!

 

[fzero]

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.