﻿Consider a smooth manifold of dimension d. This means it’s a set M that’s equipped with a topology O, with a smooth atlas A, such that it’s a smooth manifold.


If d is 1, then there is one possible choice category of smooth manifold, such that there exists a homeomorphism between each atlas in a category.


If d is 2 or 3, it’s the same story.


If d is higher than 4, then there’s a finite amount of “islands”


If d is equal to 4, there are an uncountably infinite number of islands.


This is an example of where mathematics has a direct influence on the underlying physics. It could be the case that our calculations are mistaken because we’re equipping spacetime with the wrong smooth atlas exoticness. And the only way to find the real one is to do an uncountably infinite number of experiments.


At the moment though, it seems that nearly all atlases we choose will end up making good predictions anyways.




* Maxwell equations


There are four Maxwell equations, and they can be expressed using differential forms as follows.


Basically the Maxwell equations are a set of partial differential equations, relating various tensor fields on the surface of the spacetime manifold.


Law n.1: Gauss law of electricity


Gauss’s law of electricity says that the amount of electrical flux going out of some 3d closed surface is equal to the amount of charge contained in the respective volume.


Law n.2: Gauss law of magnetism


Gauss’s law of magnetism says that the amount of magnetic flux going out of some 3d closed surface is always 0.


Practically, this law says it’s impossible to have a magnetic monopole.


Law n.3: Faraday’s law of magnetic induction


Faraday’s law of magnetic induction says that changing magnetic flux and electric field intensity are related.


Law n.4: Ampere’s law relates everything to charge flux, or in other words current.


Ampere’s law practically says that magnetic fields can be create by two things: electric current, or changing electric fields.


Maxwell’s addition (which is the time derivative part), is incredibly important because it allows the Maxwell equations to be consistent for examples other than static fields.


They also allow for the electric field to create a magnetic field, and for the magnetic field to create an electric field. This allows for “magnetic waves” to exist.


There is a really important thing to realize: the 1-form and 2-form electric and magnetic fields are respectively related to each other, through the Hodge operator, and some extra constants.


* Stokes theorem


The generalized Stokes theorem is very simple. It relates through equality for some manifold, the manifold and it’s boundary.


The most classic form it’s seen in is in the:


Int from a to b of f’ = f(b) - f(a)