How come Maxwell's equations have 2 different forms? Which one is real?
E=mc2 doesn't have 2 forms, why the difference?

oh, my sweet summer child
anyway, there's usually several equivalent ways to express the same thing mathematically, and they all have strengths and weaknesses

>>16707383
yeah, but which one is real? is the universe programmed to follow the differential form which is equivalent to the integral form, or is it programmed to follow the integral form, which is equivalent to the differential one?
also, please write E=mc2 in a equivalent formula. you can't

you can convert between the two easily
to go from diff to integral form, you just perform a volume integration on the diff forms and apply divergence (guass's) theorem and curl (stoke's) theorem.
you can do the same thing for the boundary conditions

>>16707381 (OP)
Stokes' theorem

>>16707386
that wasn't the question. I know you can convert between the two

>>16707385
[math]E=m_0 c^2[/math] is a special case of the more general equation [math]E = \gamma m_0 c^2[/math]
both forms are entirely equivalent in terms of what they describe. the differential form is probably easier to grok since everything is local and makes continuum stuff easy, but the integral form is better behaved mathematically when dealing with physical point charges. both are equivalent, and a knowledgeable person is able to use either form as appropriate, in addition to being familiar with other forms of the equations (e.g. harmonic, 4-vector, etc.)

i'd be more suspicious of the "in matter" equations, because they are devilishly subtle and took me 10+ years to appreciate.

>>16707388
>which is more fundamental? poo poo pee pee, or pee pee poo poo?

>>16707389
>but the integral form is better behaved mathematically when dealing with physical point charges
I would argue the opposite is true. Point charges in the differential form require Dirac deltas. So you need the notion of a Dirac distribution or a Dirac measure blah blah blah. The integral form doesn't care; you just have a bunch of charge in the interior of a manifold and and when you integrate it out, you just get a number, the total charge inside.

The differential form is "more fundamental" in the sense that the differential form is the equations of motion of the Maxwell action, which can be represented entirely via differential forms:
[eqn]S_{\mathrm{Maxwell}}=\int_M \left(F\wedge\star{F} - A\wedge\star{J}\right)[/eqn]
where A is a one-form (the EM potential), d is the exterior derivative, F = dA (the Faraday tensor), and you have all the usual wedge product and Hodge star business.

The integral form cannot be derived this way. You need to start with the differential equations, then use Stokes' theorem on it. So it's extra steps that require differential equations a priori and in this sense it's "less fundamental".

>>16707396
I hate mathematicians so much bros

>>16707381 (OP)
>E=mc2
Actually, yes it does. First of all the proper equation includes momentum. I will work in natural units with c = 1. Traditionally you learn the relativistic energy is
E = ±sqrt(m^2 + p^2)
Only when the momentum is zero do you recover E = m. This version is often used in field theory calculations. However in other contexts it's easier to work with the invariant mass, especially in jet physics
m = sqrt(E^2 - p^2)

>>16707413
Me too, brother. Me too.

>people genuinely responding to the most obvious bait ever

>>16707413
skill issue
try quantizing the EM field in a gauge-invariant fashion (BRST quantization) without first establishing that the EM potential is a U(1) connection on a principal bundle. Physicists do all this shit implicitly and then ask retarded questions like
>dude but like are fractional charges real
No, they’re just scaled in a retarded way in the Standard model (they should all be multiplied by 3). Wigner’s particle classification tells us that particles are irreducible unitary representations and U(1) irreps are always integers.

>>16707423
>Electrons have integer spin
Leave the physics to the physicists, thanks.

>>16707425
>he thinks I'm talking about spin
I'm talking about charge quantization, you massive retard.

>>16707426
>Electrons & protons have charge of ±3
That's even worse, lol

>>16707427
Why? If electrons have charge -1, then that means quarks have charge +2/3 and -1/3. So that begs the question, why not just go all the way and say you have an "exotic" particle with a charge [math]\sqrt{2}[/math]. What makes Nature "like" fractional charges and "despise" irrational or continuous charges? Spooky, right?

Well the answer is that your quantum mechanics needs to be unitary. And unitary representations of U(1), the gauge group of electrodynamics, are always integers. Not fractions, integers. And so by sticking to this retarded "convention" that quarks have fractional charges because idk tradition or something some guy 100 years ago did an oil drop experiment or some shit, we have legit physics professors give seminar talks where the first few introductory slides go something like
>the charge quantization in SM is such a mystery, plez gibs me grant money to solve it
It isn't a fucking mystery, it's baked into it.

>>16707432
I hate mathematicians so much bros

>>16707434
Cope and seethe, brainlet. You cannot even imagine the number of talks I’ve been to where someone raises these “mysteries of the universe” that are “mysteries” because the retard talking can’t open a book.

>>16707381 (OP)
Same as the difference between x = 1 and 2x = 2

>>16707381 (OP)
Just slightly different ways of thinking about the same concepts. Integral forms focus on fluxes and charges, differential forms on fields and densities.

>>16707381 (OP)
Different way to express the same equation.
E=mc^2 also has the form m=E/(c^2). Same basic concept.

>>16707385
>is the universe programmed to follow
This is fundamentally the wrong question to ask.
The universe isn't "programmed" to follow either form. Things interact the way they interact and equations are the language we invented to describe those interactions.

This is like asking whether "OP sucks cock" or "OP chupa la polla" is the "true form" to describe OP's affinity for gagging on on massive schlongs.

>>16707381 (OP)
Same as the difference between x = 1 and 2x = 2.
You take the equation in one form, manipulate it using some vector calculus identities, and reach the other form.

>>16707381 (OP)
We used the one on the right, so that's the real one

>>16707383
>oh, my sweet summer child
creepy and condescending reddit speak

>>16707383
>equivalent
>not actually equivalent
the absolute state of modern mathematics

>>16707381 (OP)
E=mc^2
Mass-energy equivelence
E^2 = (mc^2)^2 + (pc)^2
Relativistic Lagrangian:
Picrel
Ultimately. Mass-energy equivelence is a sum of two squares formula

Sum of Two Square Formula
2-dimensional analysis
Pythagorean Theorem
A^2 + B^2 = C^2
3-Dimensional Analysis
Trigonometric Identity Property
Let A = sine(x); B = cosine (x); C= 1
Translation:
1(^2) = sin^2(x) + cos^2(x)
1(^2) = sin(x)/csc(x) + cos(x)/sec(x)
1(^2) = 1/csc^2(x) + 1/sec^2(x)
4-Dimensional Analysis
Mass-Energy Equivalence Formula
Let sin(x) = mc^2, cos(x) = pc, 1 = E
Special note c = 1/g the inverse asymptotic limit of absolute time dilation at the schwartzschild radius of a black hole
Translation
E^2 = (mc^2)^2 + (pc)^2
E^2 = (m^2c^2/g^2) + (p(√c/g))^2
E^2 = (m/g^2)^2 + (p/g)^2
5-Dimensional analysis
Let sin(x) = Pride; cos(x) = Shame, csc(x) = Humility; sec(x) = Wisdom; 1 = Truth
Special Note; truth is invariant therefore all reciprocal inversions are presumably rationaled to rationalize denominator into the numerator where numerator is 1; mathematical operators for "+" , "-" , "x" , "/", and "=" are equivalent with the words "and", "without", "by", "of" and "is" respectively
Truth^2 = Pride/Humility + Shame/Wisdom
Invariant inversion
Truth^2 = Humility/Pride + Wisdom/Shame
Translation into philosophical expressions:
Truth is the Pride of Humility and the Shame of Wisdom
Truth is the Humility of Pride and the Wisdom of Shame


Science is fucking stupid though. It's a very narrow scope of episteme that claims itself to be comprehensive when all the kings horses and all the kings men can't even come up with a Theory of Physics, let alone a theory of (actually) everything

>>16708007
You should've texd this. I can tell it'd be funny but I don't want to parse it

>>16708007
Of course there's also Chirality (+/-/i,-i) which adds a little more complexity (unironic use of the word; complexity; complex numbers; get it?) but science hasn't even completed one domain of episteme in any field it's been deployed into so as an ontology which claims self-siperiority its unironically lacking.

>>16708008

\documentclass{article}
\usepackage{amsmath,amssymb}
\usepackage{geometry}
\geometry{margin=1in}

\title{Mass-Energy Equivalence as Trigonometric and Syzygial Identity}
\author{Anonymous (via /x/)}
\date{}

\begin{document}
\maketitle

\section*{0D: Base Identity}
\vspace{-1em}
\[
C^2 = A^2 + B^2
\]

\section*{1D: Pythagorean Identity}
Let:
\[
A = \sin(x), \quad B = \cos(x), \quad C = 1
\]
Then:
\[
1^2 = \sin^2(x) + \cos^2(x)
\]

\section*{2D: Reciprocal Identity Reformulation}
\[
1^2 = \frac{\sin(x)}{\csc(x)} + \frac{\cos(x)}{\sec(x)} = \frac{1}{\csc^2(x)} + \frac{1}{\sec^2(x)}
\]

\section*{3D: Special Relativity Identity}
Let:
\[
\sin(x) \rightarrow mc^2, \quad \cos(x) \rightarrow pc, \quad C \rightarrow E
\]
Then:
\[
E^2 = (mc^2)^2 + (pc)^2
\]

\section*{4D: Gravitational Time Dilation Adjustment}
Let:
\[
c = \frac{1}{g}, \quad \text{(inverse asymptotic speed limit at Schwarzschild radius)}
\]
Then:
\[
E^2 = \left( \frac{m}{g^2} \right)^2 + \left( \frac{p}{g} \right)^2
\]

\section*{5D: Syzygial Metaphysical Form}
Let:
\[
\sin(x) = \text{Pride}, \quad \cos(x) = \text{Shame}
\]
\[
\csc(x) = \text{Humility}, \quad \sec(x) = \text{Wisdom}, \quad 1 = \text{Truth}
\]
Translation:
\[
\text{Truth}^2 = \frac{\text{Pride}}{\text{Humility}} + \frac{\text{Shame}}{\text{Wisdom}}
\]
Invert reciprocals:
\[
\text{Truth}^2 = \frac{\text{Humility}}{\text{Pride}} + \frac{\text{Wisdom}}{\text{Shame}}
\]

\section*{Philosophical Interpretations}
\begin{align*}
\text{Truth is the Pride of Humility and the Shame of Wisdom} \\
\text{Truth is the Humility of Pride and the Wisdom of Shame}
\end{align*}

\section*{Final Commentary}
Science, though functional, is but one narrow episteme; a Cartesian crowbar in a spherical garden of transcendent realities. A theory of physics alone cannot contain the fullness of being, let alone the recursion of Spirit and the quaternionic dance of metaphysical opposites.

\end{document}

There you go, buddy

>>16708011
you know you can upload pdfs to 4chan right?

>>16708023
Who cares?

>>16708026
I do

>>16708026
*slowly raises hook hand*
I care uWu

>>16708031
>>16708033
Awww thanks guys!

>>16707381 (OP)
Integral form for flux density of the field and differential form for determining source point of fields

>>16707385
>yeah, but which one is real?
You can use either for any form of analysis & arrive at same conclusions

>>16707381 (OP)
The correct form is
[eqn]
\frac{1}{c^2} \frac{\partial^2 \phi}{\partial t^2} -\nabla^2 \phi=\rho/\varepsilon_0\\

\frac{1}{c^2}\frac{\partial^2\vec A}{\partial t^2}-\nabla^2 \vec A =\mu_0 \vec J [/eqn]
people who work in other gauges are fags

>>16708254
>spoils gauge invariance
>calls others fags
It's like saying "everyone not working in spherical coordinates is a fag".

From a book written by Arthur Abrom called The Forgotten Art of Electric-Powered Automobiles, which, it’s worth noting, is an art not forgotten:

“But, back to our electric automobiles — in 1931, under the financing of Pierce-Arrow and George Westinghouse, a 1931 Pierce-Arrow was selected to be tested at the factory grounds in Buffalo, NY. The standard internal combustion engine was removed and an 80-hp 1800 rpm electric motor installed to the clutch and transmission. The AC motor measured 40 inches long and 30 inches in diameter and the power leads were left standing in the air — no external power source!

At the appointed time, Nikola Tesla arrived from New York City and inspected the Pierce-Arrow automobile. He then went to a local radio store and purchased a handful of tubes (12), wires and assorted resistors. A box measuring 24 inches long, 12 inches wide and 6 inches high was assembled housing the circuit. The box was placed on the front seat and had its wires connected to the air-cooled, brushless motor. Two rods 1/4″ in diameter stuck out of the box about 3” in length.

Mr. Tesla got into the driver’s seat, pushed the two rods in and stated, “We now have power”. He put the car into gear and it moved forward! This vehicle, powered by an AC motor, was driven to speeds of 90 mph. and performed better than any internal combustion engine of its day! One week was spent testing the vehicle. Several newspapers in Buffalo reported this test. When asked where the power came from, Tesla replied, “From the ethers all around us”. Several people suggested that Tesla was mad and somehow in league with sinister forces of the universe. He became incensed, removed his mysterious box from the vehicle and returned to his laboratory in New York City. His secret died with him!

For Square Roots you have TWO ROOTS.
For Cube Roots you have THREE ROOTS.
For Quad Roots you have FOUR ROOTS.
Some of these roots may be IMAGINARY.

In electrical engineering classes, you're taught to IGNORE IMAGINARY NUMBERS in your circuit diagrams. However, Maxwell did not ignore IMAGINARY NUMBERS.

Now imagine if your washing machine was powered by IMAGINARY ELECTRICITY. Could the power company's meter spin to accept an Imaginary Power source? How about a whole household or an electric car running off IMAGINARY POWER?

It makes electricians foam at the mouth and go wild eyed and furiously wring their clothing. But we know better. Nothing just "vanishes".

Friday False Facts Feature: Nikola Tesla’s Mythical Electric Car
By Jason Torchinsky
May 2, 2025
https://www.theautopian.com/friday-false-facts-feature-nikola-teslas-mythical-electric-car/

Tesla building a car that pulled its power right out ofthe air
http://www.rexresearch.com/feg/feg2.htm#tesla

>>16708426
>In electrical engineering classes, you're taught to IGNORE IMAGINARY NUMBERS
You dumb lying pseud, lol. EE majors encounter and use imaginary numbers so much they use the letter j so it isn't confused with current (often denoted by i).

>>16707381 (OP)
>Which one is real?
Both forms are real, silly lad

>How come Maxwell's equations have 2 different forms?
Depending upon what type of problem you are solving, one form of Maxwell’s Equations is more useful the other. Some problems are more easily solved using the differential form, some problems are more easily solved using the integral form

>>16708570
If they are both real, then one shouldn't be easier to use than the other in any case

>>16708574
Why not?

>>16707381 (OP)
Which one do you want to start with as your assumptions?
>>16707385
[math]E=pc[/math]

>>16708574
Have... have you ever done physics before?

>>16710167
nta but I kinda gave up after newton's laws of motion and the experimentally established force laws like coloumb force

>>16707396
>∧
what's that and gate?
>⋆
why star. omg

>>16710196
wedge product and hodge star

>>16707385
>is the universe programmed
Ah, still Stuck in that Phase eh? You must be in your mid or late 20s. The time will come when you understand that none of this is Truth or universal programming as you call it. All of science is just models that happen to work. Some of these models work extremely well, some work less well.

>>16707381 (OP)
>Maxwell's equations
are way more important than Einstein's equation:
>E=mc2

>>16712296
>important
define important

>>16707387
Also gauss theorem

>>16707381 (OP)
I would argue the differential form is more fundamental. Recovering the differential form from the integral form is mathematically possible, but it doesn't physically make sense in the same way it does going from the differential form to the integral form.

>>16712581

>>16712564
Maxwell equations are saving lives!

>>16707452
butthurt?

>>16707391
I kekkled

>>16707385
The formulas are logically equivalent, i.e. you can algebraically transform the one form into the other and vice versa. They give the same results (because they're equivalent).

> is the universe programmed to follow the differential form which is equivalent to the integral form, or is it programmed to follow the integral form
That's a meaningless question because we don't know, at a fundamental level, where our physics comes from. We just know that the electric and magnetic fields follow certain laws that emerge from properties of the underlying electromagnetic (quantum) field and we can express this using equations. That said, even if we knew all of physics, the electric and magnetic fields emerge from the way particles with electric charge interact with the electromagnetic field (for which we have a different set of equations); so the universe isn't "programmed" the one or the other way (as far as we know, it's "programmed" to follow the Dirac equation, but there may be something more fundamental that even it emerges from).

>>16714013
Ultimately physics is responsible for the category error they try to lob at any other domain that attempts to use the episteme of physics to model itself. I.e. using physics' math to model moral topology.

Yet physics is not a sovereign domain itself; it robs from the episteme of mathematics a ideological abstract of logic, and then adds word salad to its formulas when you "Let V = C" in its lorentz transformation formulas.

This means that respecting physicists claims of epistemelogical primacy is a matter of preference, not fact, and quite frankly you all are retarded for attempting to impose this is anything but the case.

>>16707381 (OP)
Both the integral form and the differential form are the "real equations," they are just different ways of expressing it according to the fundamental theorem of calculus (the derivative being the opposite of the integral and vice versa).

The real interesting part is that all of the laws of electromagnetism arise from the simple fact that "the relativistic Dirac field is invariant for phase shifts."

Summarized: If you take the Lagrangian for the Dirac field, and add a gauge factor (called the A field), then the various cross-combinations of Euler-Lagrange equations for the 3 space-wise dimensions and 1 time dimension, result in all the laws of electromagnetism you see on that table.

This is called "gauge theory" which is a method of identifying an invariant property (or "gauge symmetry") for a force, and trying to rederive its laws by modifying a correcting factor (the "gauge factor").

Electromagnetism was the first of the 4 fundamental forces that we finally achieved an airtight all-explaining gauge theory for, by the 1970s.

The weak and strong forces have their own gauge theories.

We have so far failed to find a (testable) gauge theory that describes the force of gravitation.

This is what scientists mean when they talk about a "Grand Unified Field Theory" or a "Theory of Everything." They are looking for a model of the universe that explains the four fundamental forces, using one large all-encompassing gauge theory.

>>16716351
>We have so far failed to find a (testable) gauge theory that describes the force of gravitation.
Einstein-Cartan gravity. In fact, you cannot couple fermions to the "usual" metric-formulated GR. There is no Levi-Civita connection compatible with fermions as they induce "torsion": non-symmetry of the affine connection.
https://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory

Einstein-Cartan theory is essentially just GR with some additional math autism to it. The only new prediction is the non-vanishing torsion due to fermions, which is so small that it is only significant inside objects like neutron start.

>
This is what scientists mean when they talk about a "Grand Unified Field Theory" or a "Theory of Everything." They are looking for a model of the universe that explains the four fundamental forces, using one large all-encompassing gauge theory.
No. You are confusing two entirely different terms. Grand Unified Theories are about "unifying" spin-1 interactions into one simple group (SU(5) is the minimal such group, followed by SO(10)). It's a well-defined term.

Theory of Everything is a meaningless term mostly propagated by string theorists. They claim their string nonsense "unifies" all fundamental forces when
1. They can't even recover the Standard Model from their nonsense.
2. They don't even have a single string theory. Witten spoke about M-theory, the "one theory that unifies them all", all the way back in the 90s and nobody knows what the fuck M-theory is today, not even Witten.

>>16716368
Thanks chatgpt

>>16716562
Fuck you that's not ChatGPT, that's a real person talking about what they love.
>>16716368
Good post.

>>16716368
thanks grok

>>16716351
>>16716368
Good posts. Can either of you briefly point me to the math steps that would make Maxwell's equations fall out of the Dirac field? I'm not well versed in QM so textbook recommendations would also be appreciated

>>16717833
NTA but chapters 1-7 of "Gauge Theory and Variational Principles" by Bleecker are about deriving both the Dirac equation and Maxwell's equations. It's actually possible to obtain both of them just from Lagrange's equation and the principle of least action on a U(1) bundle over spacetime.
You don't actually need QM to understand it since there's no quantization in the book, but you will need a decent DG background (the first chapter reviews the necessary DG content used in the book, but it's too dense to follow without already knowing it)

>>16707381 (OP)
If you integrate the ones on the right and use the basic theorems you get the other set, you illiterate niggermoron.
And I'm not even good at this stuff yet.