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# Volume integral watch

1. Hi, how does one integrate:

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

Thanks!
2. 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.
3. 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?
4. 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).
5. aha! the origin indeed, it makes sense now, thanks!
6. But still makes 0?

If then the normal is and the divergence theorem gives

the first zero by symmetry, the second falling to zero with epsilon?
7. 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.
8. I mean that excluding the origin, the second integral on the right is arbitrarily small, so

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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