Differential forms & Maxwell equations

[Ben]

So, let us start with an arbitrary n-dimensional vector space [math]V_n[/math] over the field [math]\mathbb{R}^n[/math] which, for economies sake I will write as [math]R^n[/math]. I now define another vector space which I will write as [math]\Lambda^p(V_n)[/math], and call this the space of p-vectors. For now, I will just take this notation to be a representation of the following instructions:

 

from the set of [math]n[/math] basis vectors for [math]V_n[/math] select [math]p[/math] at a time and call this another vector space (with the usual axioms). A coupla things should be immediately obvious:

 

[math]\Lambda^1(V_n) = V_n[/math], since I am just selecting each of the basis vectors of [math]V_n[/math], one at a time. Also [math]\Lambda^0(V_n)[/math] is either meaningless (select no basis vectors!), or it is something like a set of scalars. In fact it is so defined.

 

A word of caution; here, as often elsewhere, no real distinction is made between scalars proper and scalar-valued functions. Thus we may have [math]f:R^n \to R^n,\,\, f \in \Lambda^0(V_n)[/math]. Also obvious should be that we must insist that [math]p \le n[/math].

 

An obvious question then is what is the dimension of the space [math]\Lambda^p(V_n)[/math]. Consider first [math]\Lambda^2(V_n)[/math]. Elementary combinatorics tells us that, when [math]n=3[/math] there are [math]\frac{n(n-1)}{2}[/math] choices, and we will write the dimension of this space as [math]\begin{pmatrix}3 \\ 2 \end{pmatrix}[/math] (say "from 3 choose 2").

 

Now quite obviously, this doesn't generalize, so let's write the general form as [math]\dim \Lambda^p(V_n) = \begin{pmatrix}n\\p\end{pmatrix}=\frac{n!}{p!(n-p)!}[/math] - a quick calculation shows the equivalence when [math]n=3,\,\,p=2[/math]

 

So I want to call the elements in [math]\Lambda^p(V_n)[/math] as differential p-forms, hence the elements in [math]\Lambda^1(V_n) = V_n[/math] are differential 1-forms; this requires more work (ugh!) and also, as we shall see, suggests a slight specialization for the space [math]V_n[/math].

 

Enough for now.

 

Class - pay attention!!

[Pyrotex]

Yeah. Immediately obvious. Right. Sure.

Woops, look at the time! Got to go.

[Ben]

Pyrotex: Just because this stuff is terribly familiar to you, doesn't mean that it is to all readers.

 

Or perhaps you think a "tutorial" thread is out of place here? I cannot tell. But I can say I took advice here and received encouragement.

 

Losing heart with this site - someone please cheer me up!!

[Tormod]

Hey Ben! Cheer up! ;)

 

How about explaining some of this in layman's terms (what are you describing?).

[Ben]

What's to explain, further that what I have done so far?

 

Say what: tomorrow I shall go to my local optician and get fitted up with a pair of spectacles, as I had thought I had read that this sub-forum to be a place for physics AND math discussion.

 

What did I do wrong in this context?

[UncleAl]

Lovely use of Latex (la-tech, German "h"). Are you getting enough fibration in your diet? The local average expertise spans good high school to adequate college undergrad. You could hold a fruitful conversation with John Baez. Here, show where the path leads while you lay down asphalt to get there.

 

Uncle Al is an organic chemist. Synthesize the following from any commercial starting materials and reagents. See? (including visualizing the stereograms) "8^>)

 

http://www.mazepath.com/uncleal/chiral3.gif

http://www.mazepath.com/uncleal/chiral2.gif

[Pyrotex]

Pyrotex: Just because this stuff is terribly familiar to you, doesn't mean that it is to all readers....someone please cheer me up!!

Ben! Cheer up!

The fact is, your first post went right over my punkin head, and hence my facetious reply.

The notation is familiar.

I remember vaguely what a vector space is, and could look it up in Wikipedia if I had the time.

But I don't have a clue of a clue what you're up to.

Where are you going with this?

Why did you start with assuming a vector space?

[Pyrotex]

... (including visualizing the stereograms) "8^>)

 

http://www.mazepath.com/uncleal/chiral3.gif

http://www.mazepath.com/uncleal/chiral2.gif

 

Uncle Al :D

 

I simply adore stereograms.

 

However, yours require that the viewer (me) focus the eyes "at infinity" to get the images to overlap and merge. Your 'grams are too far apart for me to do that, and I suspect, too far apart for a lot of others. The attempt gives me severe eyestrain.

 

Now - - - if you would REVERSE the two images so that the viewer can comfortably cross their eyes (focusing at a point halfway between nose and screen), then we all could appreciate your stereograms without the mild headache.

 

Pyro :hihi:

[UncleAl]

Stand further back from the screen for easier convergence. If you view the stereogram crosseyed it is front-to-back inverted showing R-configuration not S. What physics does with math chemistry does with stuff. Find the rules then bend them. Maxwell's equations are elegant! Pyotr Ufimtsev's 242 page essentially unreadable, "Metod Kraevykh Voln v Fizicheskol Teorii Difraktsii," (Moscow: Sovetskoe Radio), 1962 rocked the boat on 17 January 1991. Uncle Al will show you what can be done with stuff...

 

UNDER SATAN'S LEFT FOOT

Vote Uncle Al a big fat 10! Somebody should look.

[Pyrotex]

I love Maxwell's Equations!

Learning them in college was as close to a religious experience I've ever had.

Just noticed the title of this thread!

So, Ben, are you trying to derive Maxwell's Equations???

Then carry on. :)

 

PS: Uncle Al, I looked, and I gave you big fat "9".

There were perhaps a dozen sentences I could not understand, so I docked you one point.

:D :D :D :D :D

[UncleAl]

A 9 is good! Thank you. My proposal stands on its own feet for honest evaluation. Physical theory postulates the vacuum is isotropic (identical properties in all 4(pi) steradian directions). This can be true and *still* have a footnote.

 

Christmas 1956 Yang and Lee were pariahs for suggesting particle physics and its mirror image were experimentally different. On New Year's Day 1957 particle physics was rewritten to predict their observation: the universe is left-handed. Apologies for prior art were tendered December 1957,

 

Physics 1957

 

Somebody should look at gravitation, too, hence the entertaining UNDER SATAN'S LEFT FOOT and the serious proposal behind it. A formal paper will be published first quarter 2010 - passed Referees, waiting for galley proofs.

[Ben]

I thank Uncle Al and Pryotex for their (successful) attempts to derail this thread.

 

Unless someone can convince me there is a relation between differential forms and stereograms, and also if any totally sane person would refer to themselves in the third person singular, then I am out of here for good.

 

And no, I won't demand a refund for my subscription

[UncleAl]

You should begin with a level of expertise compatible with the crowd's abilities, then ramp up. You should state your conclusion or goal to build anticipation along the trip.

 

When you get a new paper, what do you read? You read the abstract, the conclusion, then the first paragraph. The difference betwen rape and seduction is salesmanship.

[TheBigDog]

I thank Uncle Al and Pryotex for their (successful) attempts to derail this thread.

 

Unless someone can convince me there is a relation between differential forms and stereograms, and also if any totally sane person would refer to themselves in the third person singular, then I am out of here for good.

 

And no, I won't demand a refund for my subscription

I for one might learn something from you continuing. Continuing is of course your decision.

 

Bill

[Ben]

Let me apologize to any and all members to whom my recent outburst may have caused offence. Sorry.

 

My fault was in assuming there would be some general interest in this subject, when I had absolutely no reason to make that assumption. I was also perhaps at fault for assuming a greater knowledge, generally, of abstract algebra than perhaps I should have done.

 

Sorry again

[Jay-qu]

Hi Ben,

 

I see what you are saying, but I dont see the link to Maxwell's equations. Either I missed something or you were going to continue on with the tutorial.

 

J

[Ben]

Hi Jay, yes I was planning a rather lengthy session, as it were, but got derailed. This seemed to indicate a general lack of interest in the subject.

 

OK, to be brief: physicists will be aware that the 4 Maxwell equations (in their re-casting by Oliver Heaviside) have an implicit symmetry.

 

In the language of differential forms, these 4 equations can be reduced to 2 whose symmetry is totally transparent; indeed, it is only the "supposed" absence of magnetic monopoles that prevents these equations (in this language) from being reduced to one.

 

Although not a physicist, I find this a lot of fun, but to explain it would have required readers to be familiar with differential forms, the exterior derivative and the Hodge (star) operator.

 

To explain these things had been my intention, but.........