﻿The key part to understanding the physical significance of Maxwell equations is to remember to add the Lorentz force. The Lorentz force finally uses the E electric intensity field, and the B magnetic flux field, and puts them to use. It’s a statement about the force a charged particle feels when moving/not moving in an electromagnetic field.


It says that F=q(E + v x B)


Essentially the force a particle will feel depends on the quantity of charge that particle has (the more charge, it multiplies the force). The base force (assuming charge 1), is calculated as the electric field in addition to the speed cross the magnetic flux. This essentially says that if the particle is moving parallel to an electromagnetic field in a way where it has curl, it will have a force applied to it.


So Maxwell’s equations are kind of toothless without the Lorentz force.


The Lorentz force I wrote above is in its standard vector calculus form, but can be translated into its differential forms form.


One fun thing to consider is that if you legitimately get a set initial condition, and boundary condition, you can straight up solve the Maxwell equations and predict the change in fields with time. You can predict the oscillation motion that an electromagnetic wave will have.


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If you want to find the difference between two points, you subtract them.


If you want to find the difference in area between two functions, you subtract them.


If you want to find the difference in volume between two overlapping objects in space, you subtract them.


This is a very general principle.


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There are two main kinds of group theory.


#1 is where you create groups from scratch, and prove that they’re groups using first-principles.


#2 is where you prove that all symmetries derived from a category, equipped with composition as the operator, is a group. Symmetries in this case means structure preserving maps on the category for the specific object. This is the group theory people use mostly when they say “consider the group G derived from X”.


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The chain rule in single dimensional calculus is very simple. It says that the slope function for the composition of two functions is the product of their composite functions derivatives (applied to the correct domain).


Essentially, it’s f’() * g’(), but because of the appropriate domain thing, it becomes:


f’(g(x))*g’(x)