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    Hi, how does one integrate:

     \displaystyle \int_{B(0,1)} \frac{x_k}{|x|}\ dV,\quad x = (x_1,\ldots,x_n)

    It should be easy right? I don't see how to get around the ugliness of n-dimensional spherical coördinates...

    Thanks!
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    dx or dV? Also, is x_k a vector, or the k_th coordinate of x?

    In any event, I expect symmetry is your friend here.
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    dV ...its from a backwards divergence theorem. Though if there's a bonus explanation of the difference on offer I'll be doubly happy!

    x_k's the coödinate.. so... |x| is radial, and x_k ranges from -1 to 1...its zero?
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    I'd expect it to be 0; for any point x, there's an opposite point -x and their contributions to the integral cancel out.

    The equivalent surface integral (using the divergence theorem) won't be 0.

    The reason there's a difference? Your integrand is not well behaved at the origin. (If you were to do the two integrals over a region that didn't include the origin, you would indeed get the same result).
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    aha! the origin indeed, it makes sense now, thanks!
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    But still makes 0?

    If  \Omega_\epsilon = B(0,1) - B(0,\epsilon),\ X = (\delta_{ij}|x|)_{j=1}^n then the normal is  n = \pm x/ |x| and the divergence theorem gives

     \displaystyle \int_{S^1} x_i \ d\sigma_1 = \int_{\Omega_\epsilon} x_i/|x|\ dV + \int_{S_\epsilon} x_i\ d\sigma_{\epsilon}

    the first zero by symmetry, the second falling to zero with epsilon?
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    I'm not sure what you're trying to say; the point is that the divergence theorem doesn't apply once you include the origin.

    If you know what a delta function is, you've effectively got a delta function at the origin, you're integrating to exclude the origin, but taking a limit doesn't give you the right answer for the integral, because it's totally "did you include the origin or didn't you?", which isn't the kind of thing you can take a limit over.
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    I mean that excluding the origin, the second integral on the right is arbitrarily small, so

     \int x_i\ d\sigma < \kappa,\quad\forall \kappa > 0

    and the integral must be zero.

    edit: latex isn't displaying on my phone, but I think that's just 'cause the browser is rubbish, sorry if not.
 
 
 
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Updated: October 29, 2011

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