Vector analysis and distributions

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SUMMARY

The discussion centers on the mathematical treatment of vector analysis and distributions, specifically addressing the Laplacian of the function \(\frac{1}{r}\) and its implications at the origin. It is established that \(\Delta(\frac{1}{r}) = -4\pi \delta(\vec{r})\) when \(r = 0\), indicating the presence of a delta distribution at the origin. Additionally, the divergence of \(\frac{\vec{r}}{r^3}\) is calculated to be \(-2r^{-3}\), which diverges at the origin. The use of the divergence theorem is emphasized to compute the delta distribution around the origin, confirming that the surface integral yields a constant value across different surfaces.

PREREQUISITES
  • Understanding of vector calculus, specifically divergence and Laplacian operators.
  • Familiarity with delta distributions and their properties in mathematical physics.
  • Knowledge of the divergence theorem and its applications in vector analysis.
  • Basic concepts of singularities in mathematical functions.
NEXT STEPS
  • Study the properties of delta distributions in mathematical physics.
  • Learn about the divergence theorem and its applications in various coordinate systems.
  • Explore advanced vector calculus topics, including the treatment of singularities.
  • Investigate the implications of distributions in electromagnetic theory and fluid dynamics.
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Mathematicians, physicists, and engineering students focusing on vector calculus, particularly those interested in distributions and their applications in theoretical physics.

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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