A Vector analysis and distributions

LagrangeEuler
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In many books it is just written that ##\Delta(\frac{1}{r})=0##. However it is only the case when ##r \neq 0##. In general case ##\Delta(\frac{1}{r})=-4\pi \delta(\vec{r})##. What abot ##\mbox{div}(\frac{\vec{r}}{r^3})##? What is that in case where we include also point ##0##?
 
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Direct calculation gives ##-2r^{-3}## if I do it right. it diverges at the Origin.
 
You can use the divergence theorem to compute what delta distribution it is around the origin. When you compute the surface integral you get a constant value no matter which surface you use (you can verify this with spheres pretty easily) and that implies a delta distribution at the origin. You can make this more rigorous depending on how you define a delta distribution
 
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