Exploring Singular Vector Fields & Gauss's Laws

[ShayanJ]

You're probably familiar with Gauss's laws in electricity,magnetism and gravitation:
$\oint_{\partial V} \vec{D}\cdot \vec{d\sigma}=q_V \\ \oint_{\partial V} \vec{g}\cdot\vec{d\sigma}=-4\pi G m_V\\ \oint_{\partial V} \vec{B}\cdot\vec{d\sigma}=0 \\$
It is also known that the first two integrals are non-zero because of the contributions from their singularities and the last one is zero because it is thought that it never gets singular.
Now I just feel there is something more fundamental than these from the mathematical point of view.I mean,it seems to me that there is something mathematical about singularities in vector fields that I don't know but I just can't find it or understand it myself.
I know,maybe there is nothing but my experience with mathematics tells me that it can't ignore such a...you know...mmm...anyway...just can't ignore it!
I'll appreciate any idea
thanks

 

[jvicens]

Thank you for bringing up this interesting topic. I am familiar with Gauss's laws and their significance in the fields of electricity, magnetism, and gravitation. These laws are fundamental principles that describe the behavior of these fields and have been extensively studied and verified through experiments.

Regarding your observation about the non-zero integrals of the first two laws and the zero integral of the third law, I would like to offer some insights from a mathematical perspective. The presence of singularities in the first two integrals is due to the fact that these laws describe the behavior of fields around point charges and masses, which can be considered as sources or sinks of the field. In other words, the fields have a singularity at the location of the charge or mass.

On the other hand, the third law describes the behavior of a magnetic field, which is fundamentally different from electric and gravitational fields. Unlike electric and gravitational fields, magnetic fields do not have sources or sinks, and therefore do not have any singularities. This is why the integral of the magnetic field is always zero.

From a mathematical point of view, singularities in vector fields can be better understood through the concept of distributions. A distribution is a generalized function that can represent a singularity in a mathematical expression. By using distributions, we can make sense of the non-zero integrals in the first two laws and understand their behavior around singularities.

I hope this helps to shed some light on your question. As always, in science, there is always room for further exploration and understanding. I encourage you to continue your research and exploration on this topic and share your findings with the scientific community.