3:15 PM
@ACuriousMind lol
that's a thiq dot
@BalarkaSen I have a statement that the functions of the form $h(s,t)=f(s)g(t)$ are "dense in the space of all functions of two variables in the compact $C^\infty$ topology"
Does that mean anything to you?
Sounds like some Hirsch Chapter 2 stuff to me but I'm not sure

@JohnRennie Are you around?

@BernardoMeurer Hi. What's up?

@JohnRennie Take a look at the struct branch on the github repo :)
It's looking good
The collapse function is really broken though :P
it will exit with SIGABRT

I love the overall structure ...

3:22 PM
@JohnRennie :)
But... ?

No, there's no but.

@BalarkaSen It might just be some Frechet topology with uniform convergence of all derivatives on compact sets

Though I note you have single struct/class that does both board handling and game logic.

So the usual $\mathscr C^\infty$ topology from functional analysis
who knows
what's important is that it's dense, and idk how to prove that one

I might be tempted to restrict the board struct/class to just managing cells on the board. Then use a separate struct/class for the game. That way you could reuse the board for other games.

3:25 PM
@Secret amazing, looks substantial, thx for sharing, more blurring of the QM/ classical boundary. any reaction yourself?

@BalarkaSen I can prove the products are dense in the $\mathscr C^0$ topology using locally compact Stone-Weierstrass, but how would I go about it for the $\mathscr C^\infty$ topology?

@vzn Lubos blog in one of his well answered PSE already showed that electromagnetic fields are fundamentally quantum in nature (otherwise how it can rotate the spin state of an electron), thus it is not very surprising. However, the realisation of PR boxes via beams of light does suggest that we can use lasers to investigate deeper about correlations in quantum mechanics and elsewhere
Meanwhile, I sometimes don't really like people quantifying things with just one number via extremely complicated expressions. For example, just staring at the numbers $2 \sqrt{2}$ and $4$ it is not very clear what are the difference between entangled states and PR boxes

@0celouvsky iunno. That's a nice fact tho

@Secret afaik using CHSH inequalities on classical fields is very new/ cutting edge research (only few yrs old), although this is not the 1st along those lines...

The calculations a few post back by me on CHSH stuff suggest how little I understood how to pick correct device settings
Perhaps I am unable to get the expected $2\sqrt{2}$ might be because I use the identity operator as an observable
where it has only one eigenvalue and all vectors are eigenvectors

3:34 PM
@BalarkaSen I think one has to use Stone-Weierstrass (do you know it?) for each multiderivative somehow

i do

@Secret it appears to me it is only a matter of time before someone finds a purely classical system with entirely quantum formalism associated with it... o_O ... have been musing on that myself for years...

Light is weird, I think it is best described as a relativistic quantum object, no matter how classical it seems...

@BalarkaSen I'm thinking one has to use the "crazy" version of it for function algebras, not just polynomials. If you try to approximate with polynomials you're just going to run out of derivatives eventually.

3:39 PM
Ah, you're talking about the version which says certain classes of subalgebras are dense in C^0(X) for locally compact X?

ok, it seems I get that wrong calculation because I choose the identity as an observable. The CHSH seems to need an observable with 2 eigenvalues to work, but the identity has only one eigenvalue: 1

@JohnR: Hi :-) Did u check ur mail?

@Kaumudi.H Hang on, I'll check now ...

@BalarkaSen It's dense for compact $X$, but not quite dense for just locally compact.

@Kaumudi.H Good luck, though you have massively reduced the amount of luck you need by working so hard for the last xxx months!

3:43 PM
@JohnRennie I will probably do that later tonight

you can get a sequences that converge "locally," i.e. on each compact set

Could you check the BoardCollapse function see if you spot anything fishy?

@JohnRennie Thank you :-)

but you might not have convergence everywhere

It's exploding

3:43 PM
the sup norm is badly behaved on noncompact spaces

In order to understand the more nonlocal aspects of quantum mechanics, one must understand joint probabilities
and this stuff is currently still too hard for me

@BernardoMeurer I'll have a look at the collapse function. I can't build your code because I don't have a unix box to hand, so I'll have to cobble together some test environment.

@0celouvsky Gotcha. This stuff's too hard for me.

@JohnRennie SDL is multi-platform, it should work just fine on Malwindows
:P

@Kaumudi.H here's hoping you get the right questions :-)

3:46 PM
@AccidentalFourierTransform very interesting thx much hadnt heard of that one! but its probably not purely mathematically equivalent, am guessing it is missing something key eg heisenberg uncertainty principle or something... but would like to see a close analysis of "what is missing"... think the gap could be bridged...

@JohnRennie :-) Yeah...

@BalarkaSen you might need sigma compactness too, so that you don't have "too many" compact sets

0

I was watching a show on Nikola Tesla where they stated that invention of AC current credit goes to Tesla. And patent licence was given to Westinghouse. After some year Tesla have observed loss of company he have teared the licence. I am curious to know how much money 💰 Tesla may have made if h...

Approximation theory is awful

@BernardoMeurer line 578 you used i instead of j

3:51 PM
Hi, everybody.

Danielsank, is the identity operator not an observable, despite it is hermitian?

@Secret eh?

@JohnRennie Fixed
SIGABRT
lol

You want an experiment that measures the identity operator?
The problem is that all the eigenvalues are the same, so you don't have to actually measure anything.

Measure a particle's momentum, then measure its momentum times the identity operator

3:54 PM
@JohnRennie Ran it on Valgrind
==20605== Invalid write of size 4
==20605==    at 0x401A1E: boardCollapseCells (main.c:584)
==20605==    by 0x401321: main (main.c:179)
==20605==  Address 0x7670ccc is 4 bytes before a block of size 20 alloc'd
==20605==    at 0x4C2CF35: calloc (in /usr/lib/valgrind/vgpreload_memcheck-amd64-linux.so)
==20605==    by 0x40196E: boardCollapseCells (main.c:565)
==20605==    by 0x401321: main (main.c:179)
==20605==
==20605== Invalid read of size 4
==20605==    at 0x401A07: boardCollapseCells (main.c:578)

@BernardoMeurer: this is a minor detail, but you don't initialise buf to EMPTY. Using calloc initialises it to zeros, but if you ever redefined EMPTY to anything other than zero your code would break.

@BernardoMeurer sounds like buffer overrun... so now youre writing your own malware? o_O :P
4

^ LOLOLOL

@DanielSank Ah I see. Btw the question arises when I tried to compute the CHSH correlation function for the triplet bell state (setting $\theta = \frac{3\pi}{2}$) is that it for some reason give me a number less than $2 \sqrt{2}$ instead of $2 \sqrt{2}$

rekt

3:56 PM
2 hours ago, by Secret
$$\mathcal{B}=\{\lvert++\rangle, \lvert+-\rangle, \lvert-+\rangle, \lvert--\rangle\}$$

$$\lvert \updownarrow\rangle=\frac{1}{\sqrt{2}}(\lvert +\rangle+\lvert -\rangle)$$

$$\lvert \updownarrow -\rangle=\lvert \updownarrow\rangle\otimes \lvert - \rangle$$

$$R (\theta)=\begin{pmatrix}1 & 0& 0& 0 \\ 0 & 1& 0& 0 \\ 0 & 0& \cos \theta& -\sin \theta\\ 0 & 0& \sin \theta& \cos \theta\end{pmatrix}$$

$$\lvert m\rangle=R(\theta)\lvert \updownarrow -\rangle=\frac{1}{\sqrt{2}}(\lvert +-\rangle-\sin \theta \lvert -+\rangle+\cos \theta \lvert --\rangle)$$

Dammit I got rekd
@vzn Good one :P

and I suspect the calculation fails might be because the identity operator has only one eigenvalue instead of 2 distinct eigenvalues

@JohnRennie True, indeed
@JohnRennie I suspect there's something fishy on the iterator, more than that I can't tell

@Secret I don't understand what you're trying to do.
If you want to write a detailed question and post it on main, I can take a look.

ok, I will try to clarify it on the main

3:59 PM
I would use:
for (unsigned int i = 0; i < self->size; i++) {
for (unsigned int j = 0; j < self->size; j++)
and use separate variables for the cell counters. That seems simpler to me.
@BernardoMeurer I have to go. Ping me if you haven't made any headway by the end of today and I'll have a look tomorrow morning.

@JohnRennie Alrighty, I'm working on my Computer Architecture lab today, should put more work on this tongiht :)
Thanks for all the help!

I have to say this conversation is interesting from the point of view of understanding the advantages and disadvantages of various languages.

4:13 PM
::crickets::

4:29 PM
::tumble weed blows by::

@skullpetrol I actually saw one of those yesterday.
Was eating lunch with my parents in a taco joint near my work.
A pretty big tumbleweed blew into the side of the restaurant. An employee went outside, picked it up, and brought it inside the restaurant.
She took it into the back.
I don't know if that was to use it in the tacos or because it was too big to fit in a normal trash bin.
The thing was like five feet in the long diameter.

@DanielSank hopefully the latter =P

How about trying to get Walter Lewin in here as our next guest on the AMA? @vzn @DanielSank

didn't he get fired for some sexual shit

4:37 PM
@skullpetrol all for it easier said than done... heres the place for those great suggestions plz +1 at least physics.meta.stackexchange.com/questions/9068/… ... btw saw that earlier, latest lewin user is not known to be him...

[Example of a scary integral]
$$\int_{a}^{\int f(x) dx} g(y) dy$$

@skullpetrol Go for it.
Who dat?
@Secret Heheh, I always forget how to take derivatives of stuff like that.

He's an ex MIT physics prof @DanielSank

Hey, non-USA users, here's a really nice example of very traditionally USA music:
@skullpetrol Think he'd do an AMA?

Yes @0celouvsky

4:44 PM
If the function satisfy (forgot) continuous, then fundemental theorem of calculus applies. Let G be the primitive function, then the derivative is $G'(k(x))-G'(a)$.

@DanielSank Strongly reminds of farm music, nice

@DanielSank: Hi. Exactly what is the purpose of an AMA?

@Secret Yeah, we call it country music.
@Kaumudi.H A chance to spend time asking a noteworthy person questions of interest.
For example, if you were to do an AMA, I would ask about living in India, about the educational system there, etc.

> Noteworthy

4:46 PM
"Famous"

Obviously I can do that at any time, but in an AMA there is a larger audience.

Is it OK if the person in question has been convicted of sexual harassment?

Danielsank: Nah, that's definitely NOT country music, because...
Country music sounds a lot more cliche and clone than what is shared above

@Kaumudi.H Is it ok? Well, sure, it's ok to ask criminals questions.
Do we want to host a convicted sexual harasser? Probably not.

@skullpetrol Thanks. I edited it.

4:47 PM
country music always reminds me of johnny cash

@Secret Wut? Maybe bad country music sounds cliche.

@DanielSank Right, well, this is what I meant. WL was convicted of sexual harassment.

@Secret Let $X$ be a finite-dimensional real vector space. Let $||\cdot||_1$ and $||\cdot ||_2$ be two norms on $X$. Then there exist two constants $c,C>0$ such that $\forall x\in X: c||x||_2\le ||x||_1\le C||x||_2$.

You can't define a genre by it's wort examples!

4:48 PM
@Kaumudi.H Then let's not host him.

The original man in black @BalarkaSen

^^ Good idea.

It annoys me that my timezone is opposite to anyone else, causing me need to sacrifice sleep to enjoy the party in the chat
2

@Kaumudi.H he is an amazing physics educator, no? Arguably better than Feynman was :-)

@Secret here is some "blue grass" music
It is different from "country" in the typical sounds.

4:53 PM
@skullpetrol Yes, he really is in that his lectures are great. He is not that great of a teacher for having taken advantage of one of his students.
If that sort of thing doesn't matter then do ahead with the AMA.

This is a very interesting issue.

Indeed.

It is, perhaps, a moral issue.
Then the question for me always becomes: what is the purpose of morals?
For me, "morality" means "set of behavioral patterns/rules which lead to the most productive group of organisms (i.e. us)."

MIT took down all of his lectures from their website for fear that the platform would be misused by him to contact his students and speak with them in ways no teacher should.

So then the question is whether or not having a person who is simultaneously an excellent educator and a convicted criminal would be productive.

4:55 PM
and what is the purpose of life?
on the internet

Anyhoo, I'm afraid I can't participate in what seems like a productive discussion because revision. Toodles!

Lotta people out there with lotta good qualities (great scientist, great mathematiciac, great artist, great educator) but got surprising issues with their life (marked with immoral or inethical and sometimes illegal convictions). I try not to judge them black and white, but I do not defend the crimes they are convicted with. Whether or not it's appropriate to talk to them in this chat is a different, more complicated matter admittedly
I guess I'm ambivalent about it

^ I think this is largely reasonable.
Because yes, people have all kinds of issues that we never hear about.

Agreed.

Alright, I'll be quick to add that he isn't a good educator for having taken advantage of one of his students. That matter is not a personal issue.

4:58 PM
However, having been through graduate school my instinct is to not host this person for an AMA.
@Kaumudi.H I agree with this.
Strongly.
I will attempt to explain why in a well-reasoned way.

And I will come back and read the transcript tomorrow! (:-P) Have a great day, y'all.

The relationship between professor and student is massively imbalanced. The professor holds a lot of influence over the student's future.

@Kaumudi Depends on what you mean by educator, I suppose. As a teacher, sure, I agree with this. But does that mean all of his pedagogy is of no use? Nah

Happy birthday @Kaumudi.H

It's a complicated issue, as I said. Being black and white about it is hard

4:59 PM
@yuggib Is $C^\infty(\Bbb R)\otimes C^\infty(\Bbb R)$ dense in $C^\infty(\Bbb R^2)$ in the $\mathscr C^\infty$ topology? (This is the topology of uniform convergence for each multi-derivative on compact sets.) I can't find it online. There's a sketch of this on Math Stack for compactly supported functions, but I don't think the proof generalizes.

@skullpetrol x'D Thank you.

@BalarkaSen This is absolutely true. A person's contributions are not invalidated by their failures.
Still, I think the issue of abuse in academic situations is such a big problem that it's important to push back against it very strongly.

Perhaps a meta question?

Yeah, that's what I meant. Thanks for writing that out clearly.

^ That would be very, very interesting.

5:01 PM

I am a six foot tall male and even I felt abused in graduate school.

Manlet

@BalarkaSen Please continue! This is interesting and important.
I can hardly imagine what it is like for females to deal with a sexually aggressive professor.

@skullpetrol Perhaps, but frankly I don't like that site.

5:03 PM
Me neither.

@yuggib The issue is that one can't use Stone-Weierstrass to get the right kind of convergence. One can get uniform convergence in each compact set, but eventually one needs to fix one compact set for a construction. Approximating higher derivatives is possible, but I don't see how to do it without first fixing a compact set...you have to do some integrals.
@DanielSank Why not?

My point is simply this: sexual offence by a professor against a student is part of two hugely destructive problems: the dominance of professors over their students and the dominance of men over women. I am not sure what is the right way to fight against these problems.
One can argue either way:
1) We must communicate with offenders and victims to understand what happened so we can understand how to prevent it in the future.
2) We must not tolerate this kind of behavior at all; ostracizing offenders creates a more productive setting for everyone.
I'm not sure which I believe, and so I would tend to take a conservative approach and not host the person.
Now that said, in most aspects of life, I think #1 is more true.

#2 sounds unrealistic imho

I think that pushing offenders into obscurity may cut us off from avenues of understanding that can lead to improvement.
Another point: I think it's important to find out what other users think of this person. Would e.g. women find his mere presence objectionable? I simply don't know, although @Kaumudi.H's comments make me think that she would.

What exactly did he do?

5:09 PM
For me, I will go for 1, if that does not work, 2 has to be enforced. So it's more like a stepwise management, and best to be dealt on a case by case basis as long resources available

Never mind then.

@0celouvsky y u so lazy?

I think knowing the fact that they are involved in these kind of offenses doesn't stop me from talking to them about physics or physics education (or alike) but I probably wouldn't engage with them casually or too friendly
Just sayin'

It was "ugly."

5:09 PM
@BalarkaSen Of course. This makes sense.

@DanielSank I'm doing other things.

So if this AMA is an official thing there should be no problem. If it's an informal thing maybe it's best not be done?

@0celouvsky Ah, the opposite of lazy.
@BalarkaSen Hmmm, I'm not sure I agree with that reasoning.

I'm not sure I understand the reasoning

I think the practical questions here are 1) Would users find this objectionable, and 2) Do we think an AMA with this person could happen without digressing into useless fight about his harassment offense.

5:11 PM
That's 'cuz there isn't a reasoning.

in any case, I dont really find a WL AMA interesting
Id rather have a second heather AMA
:-P

We could label discussion of his offence as "off topic"
The man is guilty
Convicted

In a court of law?

in a court of love

Let's talk physics.

5:13 PM
Those are the worst

Anyway, I'm off. I think it's best to get a non-controversial person in this AMA. This chat has had enough lulz

Cya

@skullpetrol ok, how would I approximate a $C^\infty$ function on $\Bbb R^2$ by a product of $C^\infty$ functions on $\Bbb R$

define approximate

5:15 PM
@0celouvsky Why should we care?

@AccidentalFourierTransform I want a sequence in $C^\infty(\Bbb R)\cdot C^\infty(\Bbb R)$ (product associated with pointwise multiplication) that converges in $\mathscr C^\infty$ to $f\in C^\infty(\Bbb R^2)$
@DanielSank I don't expect you to be able to answer, so you shouldn't care.

for an arbitrary $f$?
are you sure that's possible?

P. Michor made the claim, and he's a pretty trustworthy guy

On paper :P

@0celouvsky Just wondering, I suspect this is an unfinished sentence?

5:19 PM
@AccidentalFourierTransform Oh, wait, I want density of $C^\infty(\Bbb R)\otimes C^\infty(\Bbb R)$
@Secret I can show you the proof if you wish

ok

So linear combinations of functions in $C^\infty(R)\cdot C^\infty(R)$

oh that changes everything

That's what your gf told me :P

5:36 PM
Let's start small with these approximations
Clearly I'm going to be using some form of Stone Weierstrass
Suppose $\phi:R \to R$ is smooth with subpport $\subset [a,b]$. Can I find a sequence of polynomials $P_n:[a,b]\to R$ with $||P_n-\phi||\to 0$ and $||P_n'-\phi'||\to 0$, where $||f||:=\sup_{[a,b]}|f|$?

↑ re all the AMA talk ... great to see further/ continued interest. however at this juncture "our" main challenge is getting anyone to agree. can use any help available with that. even with WLs controversial bkg, dont think would personally turn him down. anyway this was triggered by a new user named WL but is likely not him, so its all hypothetical anyway. (and maybe thats why its so much more fun to discuss at extended length...) also 2nd idea of Kaumudi as guest! :)

Clearly we can find polynomials $L_n,Q_n$ such that $L_n\to\phi$ uniformly and $Q_n\to \phi'$ uniformly
Maybe I just want to integrate the $Q_n$'s
Well, $\phi$ is definitely the integral of its derivative since it's smooth
Aha.
If I write $R_n(x)=\int_a^x Q_n(t)\, dt$, then $|\phi(x)-R_n(x)|\le \int_a^x|\phi'(t)-Q_n(t)|dt\le (b-a)||\phi'-Q_n||\to 0$

I wish I could give a second downvote for the little picture thingies. — WillO 4 mins ago
😂😂😂😂

in theory salon, 39 secs ago, by vzn
→ rumor of a machine learning prj on standard model particle unification by sebastian thrun/ new yorker mag https://rjlipton.wordpress.com/2017/04/01/the-end-of-human-science/

Trying to approximate a smooth function on a rectangle with polynomials
It's hard

5:58 PM
Why people love to cramp so much information on one single number, it removes all the mechanisms on how to explain things

@AccidentalFourierTransform =P
reading the above transcript, I didn't realize what Walter Lewin had done - while I respect the work he's done teaching and working on physics, I think the fact he's done what he's done significantly deters me from even watching his lectures, let alone inviting him for an AMA.

Maybe I should be using a partition of unity
$f=\sum f_n$ where each $f_n$ is a test function
approximate each $f_n$ in $\mathscr C^\infty$
then somehow make everything work

Come on @heather he's a harmless old man now.

omg is @skullpetrol actually WL??

-_-

6:07 PM
$$\large{\sum_{y\in\int^{\int^{\int^{\int^{\int^{\int^{\int \sin (x_{7})dx_7}dx_6}dx_5}dx_4}dx_3}dx_2}dx_1}}\prod_{\prod_{\prod_{i=1}^ni}}e^{y}$$
I really need a more systematic way to handle iterative nonlocal functionals like these

wtf is that
Hmm...can I metrize $\mathscr C^\infty(\Bbb R^2)$?
Sure can
lol
the proof is in some Austrian master's thesis

6:29 PM
11

The news that MathJax CDN shutting down on April 30, 2017 was recently brought to my attention. I am testing an updated version of ChatJax which uses the alternate server given in the MathJax post. It seems to work well, but if StackExchange is going to host a local copy to use with the sites t...

@ACuriousMind
2 hours ago, by 0celouvsky
@yuggib Is $C^\infty(\Bbb R)\otimes C^\infty(\Bbb R)$ dense in $C^\infty(\Bbb R^2)$ in the $\mathscr C^\infty$ topology? (This is the topology of uniform convergence for each multi-derivative on compact sets.) I can't find it online. There's a sketch of this on Math Stack for compactly supported functions, but I don't think the proof generalizes.

That's probably an interesting question for someone who cares about such things :P

Sigh...
@ACuriousMind Would you happen to know if Universitat Wien has theses from ~1990 online?

No idea

@ACuriousMind I need to get ahold of the thesis. Is it better to email the Librarian there or the author?

6:39 PM
Also no idea, I never needed to do that

I couldn't find it on their internal search.

Try a librarian first.

If yugibb can't answer my question then I'll ask some profs at my school
If all else fails I'll just email the author of the book
He's a regular on MO, should be open to emails
Maybe
I just wish I could read this thesis, maybe it would explain this cryptic comment
What if you're Job Smith XXVI
how do you put that on forms?

What?

7:07 PM

7:42 PM
Hi. Have you seen my question? Why $\textbf{E}\bullet \textbf{B}=\textbf{E}’\bullet \textbf{B}’$?
After

There are distinct possibilities:

$\textbf{E}=\textbf{B}=0$

$\textbf{E}\bullet \textbf{B}\ne 0$

$\textbf{E}\bullet \textbf{B}\ne 0$

$\textbf{E}\bullet \textbf{B}= 0 \longrightarrow \textbf{E}’\bullet \textbf{B}’= 0$ with $\textbf{E}, \textbf{B}\ne 0$.

What is happen in these cases?

After why, when, $E^2-c^2B^2=0$ is the case of the transversal waves for any referment? There are other case? Perhaps electromagnetic plane wave? With $E^2-c^2B^2>0$ and $E^2-c^2B^2<0$ what is happen? H

please use $\cdot$ for the dot product
that fat dot is silly

(removed)
(removed)

Hi. Have you seen my question? Why $\textbf{E}\cdot \textbf{B}=\textbf{E}’\cdot \textbf{B}’$?
After

There are distinct possibilities:

$\textbf{E}=\textbf{B}=0$

$\textbf{E}\cdot \textbf{B}\ne 0$

$\textbf{E}\cdot \textbf{B}\ne 0$

$\textbf{E}\cdot \textbf{B}= 0 \longrightarrow \textbf{E}’\cdot \textbf{B}’= 0$ with $\textbf{E}, \textbf{B}\ne 0$.

What is happen in these cases?

After why, when, $E^2-c^2B^2=0$ is the case of the transversal waves for any referment? There are other case? Perhaps electromagnetic plane wave? With $E^2-c^2B^2>0$ and $E^2-c^2B^2<0$ what is happen? How shall I p
@0celouvsky done

merci

Are you racist against fat dots?

7:49 PM
yup
can't stand their general face shape

@0celouvsky :) for your opinion how can I put well my question to be put on the site?

@Sebastiano Uh, what exactly is your question?
What is $E'$ and $B'$?

electric field e magnetic field

I mean how are they related to $E$ and $B$
what's the context?

@Qmechanic good evening

7:51 PM
are they free solutions or is there a source

@0celouvsky the source is special relativity. the question are lorentz' invariants

@Sebastiano ok, so $E'$ and $B'$ are Lorentz transformed fields?
Do you have an explicit formula for $E'$ and $B'$?
I know such a thing exists

@0celouvsky yes- i am a teacher of maths and physics 42 years. in this period i am following a special course- my professor is very fast during the lesson.

The theory of special relativity plays an important role in the modern theory of classical electromagnetism. First of all, it gives formulas for how electromagnetic objects, in particular the electric and magnetic fields, are altered under a Lorentz transformation from one inertial frame of reference to another. Secondly, it sheds light on the relationship between electricity and magnetism, showing that frame of reference determines if an observation follows electrostatic or magnetic laws. Third, it motivates a compact and convenient notation for the laws of electromagnetism, namely the "manifestly...
I think that should help

@0celouvsky but there is not my exact question. anyway thank you very much. See quora.com/Why-is-this-invariant-in-Relativity-E-2-c-2B-2