Maxwell's equations are so cool!

[Anonymous No.10887337]

I like Maxwell's equations! they basically control 99,99% of our life, for example I can write this post only thanks to these equations

Also there's so many different ways to write them! What's your opinion on them?

[Anonymous No.10887340]

>>10887337
>What's your opinion on them?
They're classical equations and everything you just said is false. You cannot design modern ICs using only Maxwell.

[Anonymous No.10887352]

>>10887337
Brainlet here. Can you break down the equation and explain what the terms mean, genuinely interested in learning more about them.

[Anonymous No.10887365]

>>10887340
Maxwell's equations are used to describe the electromagnetic field. Computers use electromagnetic fields. If we exclude computer parts that make use of quantum effects, then yes we can use maxwell equations, it will be extremely retarded but you can.

[Anonymous No.10887515]

>>10887352
Alright, i'm not a scientist, but i'll try: those are the relativistic version of Maxwell's equations

The F is the electromagnetic tensor. A tensor is basically a matrix. The components of F are shown in pic related where E and B are the electric and magnetic fields, c is the speed of light.

The tensor has the propriety that it transforms according to Lorentz transformations. That means, let's suppose you measure an electric field E and no magnetic field B. Then, you start moving at a certain speed, you will find that for example that there is a certain electric field E but also a magnetic field B. So the electric and magnetic fields are not constant but depend on the reference frame (that's why iy's relativistic).

The index mu sigma nu can vary from 0 to 3 so the first equation actually represent 64 different equations, but most of those reduce to useless equations like 0 = 0. Only 4 of them are important. Those are the homogenous equations, which means that they describe how the electric and magnetic fields vary regardless if there are currents or electric charges. These laws says for example that there can't be a magnetic monopole.

The second equation (it's actually 4) describes how the the magnetic and electric fields vary when there are currents and charges. J is the four-vector called four-current. A four-vector is just a vecotr but with 4 components. The first is the charge density and the other three are the current density. A four-vector also follows the Lorentz transformation, which means that currents and charge are not constant but depend on the reference frame

[Anonymous No.10887640]

>>10887337
Maxwell's equations are pretty based. if you want to be l33t and learn quantum field theory, that's where you need to start

[Anonymous No.10887864]

>>10887515
> A tensor is basically a matrix.
No.

[Anonymous No.10887909]

>>10887864
I said "basically" to keep the explanation simple, he can imagine it as a matrix and it makes no difference in this example

Actually not even i know th exact mathematical formalism for tensors, I'm expecially confused by the lowering of indexes, why do we need to keep track of that when the difference is just one minus sign in the temporal component?

[Anonymous No.10887940]

[math] d \star F = \mu_0 J [/math]

It's been a while, but this looks nice.

[Anonymous No.10888054]

>>10887940
The exterior calculus is the superior calculus. The fire rises, brother.

[Anonymous No.10888059]

>>10887365
>If we exclude computer parts that make use of quantum effects, then yes we can use maxwell equations, it will be extremely retarded but you can.
wut

[Anonymous No.10888082]

>>10887515
Do you really call the first one part of maxwells equations? Cause that one holds up for any antisymmetric 2-Tensor doesnt it? Its just a useful identity but doesnt really tell you anything about fields.
The second one is what I was taught as maxwells equations, and it (or they) actually has to be derived using least action principle and variational calculus and heavily depends that for example [math]F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu[/math] which already puts all the relevant information about the fields into the equation and doesnt work without them.
Really just curious since I know that theres a lot of ways to do electrodynamics and maybe theres a reason you learned it like that.

[Anonymous No.10888086]

>>10888082
>>10887515
Wait nevermind Im stupid and missed the paragraph where you explain it and of course the first one also relies on the relation between F and A...
Thats interesting cause I dont think we learned that. There was just an exercise where you first had to prove that its true and then use it to prove something else...

[Anonymous No.10888169]

>>10887909
The math definitions of tensors are much more intuitive to me than physicist's have ever been.

Sticking to just contravariant tensors, a k-tensor on a (real) vector space V is just a k-linear map from k copies of V to R. Usually, V is the tangent space of a manifold at a point. You can easily construct a vector bundle out of the k-tensors at each point just by throwing them all into a bin and topologizing in the obvious way imposed by the charts. A smooth k-tensor on a manifold is then just a smooth section of this bundle.

This makes sense: given a smooth k-tensor, we can take any point x on the manifold M and k tangent vectors there and get back a number, which is what we want. Moreover, this functions varies smoothly with each of the k vectors and over M.

You can do the same to define covariant tensors using the dual space.

[Anonymous No.10888204]

>>10887337
You're the kind of person that needs to be dragged back behind the barn and executed in the middle of the night. All redditers get the rope. This is no longer a board of peace.

[Anonymous No.10888214]

>>10888169
If you don’t speak math, the definition is basically a statement of what you want:

a) should spit out numbers based on the point you are at and some collection of tangent vectors or linear functionals on the tangent space at that point, varying nicely. This is basically the point of a tensor.
b) should it depend on tangent vectors or linear functionals, it should be linear in each argument, so that it has some kind of formal distributive property that make it amenable for describing properties based on lengths, areas, volumes, etc.
c) allowing yourself both co- and contravariant tensors (and mixed tensors) not only allows you to input both vectors and functionals, but you can also do “currying” tricks so that you can think of your tensor as spitting out tangent vectors or functionals instead of just numbers.

[Anonymous No.10888250]

>>10888169
explain like im 5 pls
> u can't, cause u just regurgitate definitions without true understanding

[Anonymous No.10888281]

>>10888250

>Explain mathematical abstractions to me without assuming any background in mathematics at all! Use only analogies to everyday occurrences that a child would be expected to have exposure to!

When people do this you get bullshit popsci and quantum woo. If you don't know what a vector space is, there is no explanation in the world that will give you and adequate understanding of tensors in a mathematical context. Why do brainlets insist on being spoonfed information that's been dumbed down to the point of being wrong? What's true understanding, do you want him to draw you a fucking picture?

[Anonymous No.10888284]

>>10888281
spoiled brat zoomed disease. they require all information to be reduced to tweetable memes

[Anonymous No.10889186]

>>10887337
The Abelian Yang-Mills [math]U(1)[/math] gauge theory, describing the quantum dynamics of EM, was the first instance in which Atiyah-Bott localization was applied to moduli spaces of connections. Looking back it's quite trivial but it did lay the groundwork for e.g. non-Abelian localization and Hermitian Yang-Mills theory.

[Anonymous No.10889195]

>>10887337
>calling something from science cool

confirmed popscitard

[Anonymous No.10889215]

>>10887940
I think you dropped this?
[eqn]d F = 0[/eqn]

[Anonymous No.10889239]

>>10887365
maxwell's equations aren't used to design computers
Ohm's law is

[Anonymous No.10889415]

>>10889239
Only Ohm's law? Not also Kirchoff?

[Anonymous No.10889944]

>>10889239
retarded
Microfab with high clock speeds behaves like transmission line

[Anonymous No.10889966]

>>10889186
can you give me a link? i'd like to see some of these more advanced mathy things in the context of normal-ass classical physics

[Anonymous No.10890142]

>>10889966
not him, but
http://www.damtp.cam.ac.uk/user/tong/gaugetheory/gt.pdf
specifically the stuff about non-abelian gauge theories
and also as preliminary reading http://phyweb.lbl.gov/~rncahn/www/liealgebras/texall.pdf