﻿One application of differential forms is to Maxwell equations. And I will try to explain that here.


No matter what kind of mathematical structure you use to describe Maxwell’s equations, if you want to do a practical calculation, you must always unwind the mathematical tools until you arrive at a description of the components of the electromagnetic field in a specific coordinate system. Basically, calculations require coordinates.


The reason differential form analysis is the same as vector analysis is because you have a metric, and you can turn a vector into a covector, or vice versa, easily using the metric (same with inner product).


What are the Maxwell equations and what is electrodynamics? Basically you have a few important fields, related to each other by the Maxwell equations. Considering you have an initial state, everything else can be predicted from it.


What are the important fields?


They’re categorized on three(ish) levels, and one for electric and another for magnetic. Except that the magnetic level 3 (density) is not used. And also that you have an extra one on level 2, which is the current density (which measures the actual movement of current through a surface, as opposed to electric flux density, which simply measures the effect of a charge at some distance on a surface).


First is the intensity field. You have the electric field intensity, and magnetic field intensity. They’re both 1-forms, which means you can imagine it as a series of planes that measure paths that particles take.


Second is the flux fields. You have Magnetic flux and electric flux, and also appended is flux current. They both are 2-forms, which means you can imagine them as a series of pipes. When asked about a specific surface, you can see how many pipes cross the surface to get the measurement.


Third is electric density, which is a 3-form. This means you can imagine it as a bunch of cubes in space. When given a volume, you can ask how many cubes are in the volume, and that's your measurement.


All the fields are given letters as well:
* Electric field intensity is the letter E
* Magnetic field intensity is the letter H
* Electric flux is the letter D
* Magnetic flux is the letter B
* Current flux is the letter J
* Electric density is the letter ρ


The Maxwell equations relate all these fields together using path and volume integrals.


  

Faraday’s law on induction
Ampere's law
Gauss law for electricity
Gauss law for magnetism
(in order)


Faraday’s law says that for some loop that a particle takes, which forms a path P and a surface A, it's the case that the number of planes crossed by the path is equal to the negative time derivative of the number of tubes crossed by the area. One consequence is that if the magnetic flux is constant with time, then the loop must cross a total of 0 planes (when going forward and backwards). It basically means that changing the magnetic flux, for example by moving a magnet, will cause a change in the flux through a surface, which means the panels crossed by the loop is equal to the amount of change of flux.


Ampere’s law says that in a system that is not changing with time, the surface integral of the current flux is equal to the integral of the magnetic field intensity of the path. In geometric terms that means whenever you have a current flowing through a wire, you will have a magnetic field circulating around the path outside the pipes. Ampere’s law also says that whenever you have a changing electric flux with time, you also get a magnetic field on the path, based on the rate of change with time.


Gauss’ law on electricity says that cubes spawn pipes. In other words if you have electric density in a volume, it will cause an equal amount of pipes to come out of the surface of the volume.


Gauss’ law on magnetism says that for any closed surface that you choose, the number of pipes going in and going out for the magnetic field flux is equal. So their difference is 0.


In R^3, the hodge star is a way of going from the 0-forms to the 3-forms, and from the 1-forms to the 2-forms. Geometrically it’s exactly the equality you expect.