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First, we will compute $\text{div}\,F$. The partial derivatives are given by$${{\partial F}\over{\partial x_i}} = q {\partial\over{\partial x_i}}\left({{x_i}\over{r^3}}\right) = q\left({1\over{r^3}} - {{3r^2 {{x_i}\over{r}} x_i}\over{r^6}}\right) = 0.$$Thus, $\text{div}\,F = 0$ away from the origin. Consider now a ball $B$ of radius $r$ centered at the ...


3

I have heard three theories for how space-time is shaped, flat, sphere-like, or saddle-like. Flat is the most likely, as all our measurements implies that space time has curvature close to 0. Inflation makes it so that a sphere like or saddle like spacetime evolves into a sphere like or saddle like spacetime that has a curvature very very close to zero. ...


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Is it plausible for spacetime to be shaped something like a torus? I can't give a factual answer, so I will give an opinion. People have proposed that space or spacetime has a toroidal topology. This goes back to the old asteroids game, and there's plenty of papers on the arXiv. But there's nothing actually plausible about these proposals. There's no ...


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Topological boundary modes developed in topological insulators have also been discovered in classical systems including photonic crystals and several mechanical systems. For your interest in classical mechanics, here are some recent papers: Topological protection: Of bagels and Burgers Topological boundary modes in isostatic lattices Nonlinear conduction ...


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Classically, spinors are insignificant, and should not be expected to fulfill any physical role, since they are not proper representations of the rotation group $\mathrm{SO}(3)$ or the Lorentz group $\mathrm{SO}(1,3)$. The importance of spinors does not arise from any classically intuitive thinking about orientations. Instead, the reason why spinors appear ...



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