﻿The problem is that I don’t understand the chain rule. I’m going to explain the single variable calculus chain rule for practice.


The chain rule in a single dimension says that (fg)’ = f’(g(x))*g’(x)


The chain rule in multidimensional setup is: gah I don’t understand.


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What is topological convergence?


You talk about the continuity of functions, but the convergence of sequences.


What is a sequence? A sequence is a map from N to M that essentially associates a point to each element of your series.


What is a series? A series is the sequence of the partial sum of another sequence.


When somebody talks of the sum of a series, they mean taking the limit of the series.


What does it mean for a sequence to converge? The intuitive idea is that no matter how far off an element you pick in the sequence, it should sort of stop moving and reach a solid point.


This is rigorously defined by saying that for a candidate point, for any open set that contains the candidate point, there should always be a high enough value of n such that the nth element of the sequence is contained in the open set.


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With a topological space, you can do the forward argument and find out why a continuous map is defined as being a map where the preimages of open sets are open.


You would initially expect this to be identical to the requirement that the image of any open set is an open set, but this is in fact a different restriction on f, called an open map. They are not equivalent for a simple reason: there isn’t guarantee that the image of every open set in the domain hits every open set in the target. This would be a catastrophe for continuity because then you could find an example of an open set in the target whose preimage is not in the domain.


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More generally, you can always figure out what the one direction restriction is, vs the two direction restriction. Generally it is just adding the bijective to the requirement. For example, a linear map becomes a linear isomorphism. Or a continuous map becomes a homeomorphism. For sets, any old map without restriction becomes a bijection. I bet you could do the same thing with groups.


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Maxwell equations are:
* Gauss law for electricity,
* Gauss law for magnetism
* Faraday law of induction
* Ampere’s law


The fields at play are:
* Electric field intensity (1 form)
* Magnetic field intensity (1 form)
* Electric flux field (2 form)
* Magnetic flux field (2 form)
* Electric current flux (2 form)
* Electric density (3 form)


Gauss law for magnetism says that for any closed volume V with a surface S, the number of pipes going in and out of the surface S sum up to zero.


Gauss law for electricity says that for any closed volume V with a surface S, the number of pipes going out of the surface S is equal to the number of cubes of electric density in the volume V.


Faraday’s law of magnetic induction says that for any path P with a surface A, the number of surfaces pierced by the path P is equal to negative the change in pipes going through the surface A


Ampere’s law says that when the charge doing through a wire (electric