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I'm trying to show that $$u(x)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{q(y)(x-y)}{|x-y|^3}dy$$ is conservative. By formal computation, I can say that $\nabla F=u$ for $$F(x)=-\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{q(y)}{|x-y|}dy.$$ And then I need to justify this.

The paper that I am reading says it is sufficient to show that $$\lim_{r\rightarrow 0^+}\int_{B(x,r)}\frac{q(y)(x-y)}{|x-y|^3}dy=0.$$ Where does this sufficient condition come from? I am looking at statements of the differentiation under integral sign theorem from http://planetmath.org/DifferentiationUnderIntegralSign.html, but am not seeing how it is being applied.

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What is known about $q$? –  1015 Feb 15 '13 at 1:37
I agree that this looks strange. The last property automatically holds whenever the integral is absolutely convergent. Maybe the point is in assumptions on $q$, as julien points out. –  Giuseppe Negro Feb 15 '13 at 1:52
Oops, sorry, $q\in C^1_c$. So is compactly supported and continuously differentiable. –  Stuart Feb 15 '13 at 2:18
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