This Week's Finds in Mathematical Physics (Week 8)
john baez
I was delighted to find that Louis Kauffman wants to speak at the
workshop at UCR on knots and quantum gravity; he'll be talking
on "Temperley Lieb recoupling theory and quantum invariants of links and
manifolds". His books
On knots, by Louis H. Kauffman, Princeton, N.J., Princeton University
Press, 1987 (Annals of Mathematics Studies, no. 115)
and more recently
Knots and physics, by Louis H. Kauffman, Teaneck, NJ, World Scientific
Press, 1991 (K & E Series on Knots and Everything, vol. 1)
are a lot of fun to read, and convinced me to turn my energies
towards the intersection of knot theory and mathematical physics. As
you can see by the title of the series he's editing, he is a true
believer the deep significance of knot theory. This was true even
before the Jones polynomial hit the mathematical physics scene, so he
was well-placed to discover the relationship between the Jones
polynomial (and other new knot invariants) and statistical mechanics,
which seems to be what won him his fame. He is now the editor of a
journal, "Journal of knot theory and its ramifications."
He sent me a packet of articles and preprints which I will briefly
discuss. If you read *any* of the stuff below, *please* read the
delightful reformulation of the 4-color theorem in terms of cross
products that he discovered! I am strongly tempted to assign it to my
linear algebra class for homework....
1. Map coloring and the vector cross product, by Louis Kauffman, J. Comb.
Theory B, 48 (1990) 45.
Map coloring, 1-deformed spin networks, and Turaev-Viro
invariants for 3-manifolds, by Louis Kauffman, Int. Jour. of Mod. Phys.
B, 6 (1992) 1765 - 1794.
An algebraic approach to the planar colouring problem, by Louis Kauffman
and H. Saleur, Yale University preprint YCTP-P27-91, November 8, 1991.
As we all know, the usual cross product of vectors in R^3 is not
associative, so the following theorem is slightly interesting:
Theorem: Consider any two bracketings of a product of any finite number
of vectors, e.g.:
L = a x (b x ((c x d) x e) and R = ((a x b) x c) x (d x e)
Let i, j, and k be the usual canonical basis for R^3:
i = (1,0,0) j = (0,1,0) k = (0,0,1).
Then we may assign a,b,c,... values taken from {i,j,k} in such a way
that L = R and both are nonzero.
But what's really interesting is:
Meta-Theorem: The above proposition is equivalent to the 4-color
theorem. Recall that this theorem says that any map on the plane may be
colored with 4 colors in such a way that no two regions with the same
color share a border (an edge).
What I mean here is that the only way known to prove this Theorem is to
deduce it from the 4-color theorem, and conversely, any proof of this
Theorem would easily give a proof of the 4-color theorem! As you all
probably know, the 4-color theorem was a difficult conjecture that
resisted proof for about a century before succumbing to a computer-based
proof require the consideration of many, many special cases:
Every planar map is four colorable, by K. I. Appel and W. Haken, Bull.
Amer. Math. Soc. 82 (1976) 711.
So the Theorem above may be regarded as a *profoundly* subtle result
about the "associativity" of the cross product!
Of course, I hope you all rush out now and find out how this Theorem is
equivalent to the 4-color theorem. For starters, let me note that it
uses a result of Tait: first, to prove the 4-color theorem it's enough
to prove it for maps where only 3 countries meet at each vertex (since
one can stick in a little new country at each vertex), and second,
4-coloring such a map is equivalent to coloring the *edges* with 3
colors in such a way that each vertex has edges of all 3 colors
adjoining it. The 3 colors correspond to i, j, and k!
Kauffman and Saleur (the latter a physicist) come up with another algebraic
formulation of the 4-color theorem in terms of the Temperley-Lieb
algebra. The Temperley-Lieb algebra TL_n is a cute algebra with
generators e_1, ..., e_{n-1} and relations that depend on a constant d
called the "loop value":
e_i^2 = de_i
e_i e_{i+1} e_i = e_i
e_i e_{i-1} e_i = e_i
e_i e_j = e_j e_i for |i-j| > 1.
The point of it becomes clear if we draw the e_i as tangles on n
strands. Let's take n = 3 to keep life simple. Then e_1 is
\ / |
\/ |
|
/\ |
/ \ |
while e_2 is
| \ /
| \/
|
| /\
| / \
In general, e_i "folds over" the ith and (i+1)st strands. Note that if
we square e_i we get a loop - e.g., e_1 squared is
\ / |
\/ |
|
/\ |
/ \ |
\ / |
\/ |
|
/\ |
/ \ |
Here we are using the usual product of tangles (see the article "tangles"
in the collection of my expository posts, which can be obtained in a
manner described at the end of this post). Now the rule in
Temperley-Lieb land is that we can get rid of a loop if we multiply by
the loop value d; that is, the loop "equals" d. So e_1 squared is just
d times
\ / |
\/ |
|
|
|
|
|
|
/\ |
/ \ |
which - since we are doing topology - is the same as e_1. That's why
e_i^2 = de_i.
The other relations are even more obvious. For example, e_1 e_2 e_1 is
just
\ / |
\/ |
|
/\ |
/ \ |
| \ /
| \/
|
| /\
| / \
\ / |
\/ |
|
/\ |
/ \ |
which, since we are doing topology, is just e_1! Similarly, e_2 e_1 e_2
= e_1, and e_i and e_j commute if they are far enough away to keep from
running into each other.
As an exercise for combinatorists: figure out the dimension of TL_n.
Okay, very cute, one might say, but so what? Well, this algebra was
actually first discovered in statistical mechanics, when Temperley and
Lieb were solving a 2-dimensional problem:
Relations between the `percolation' and `coloring' problem and other
graph-theoretical problems associated with tregular planar lattices:
some exact results on the `percolation' problem, by H. N. V. Temperley
and E. H. Lieb, Proc. Roy. Soc. Lond. A 322 (1971), 251 - 280.
It gained a lot more fame when it appeared as the explanation
for the Jones polynomial invariant of knots - although Jones had been
using it not for knot theory, but in the study of von Neumann algebras,
and the Jones polynomial was just an unexpected spinoff. Its importance
in knot theory comes from the fact that it is a quotient of the group
algebra of the braid group (as explained in "Knots and Physics").
It has also found a lot of other applications; for example, I've used it in
my paper on quantum gravity and the algebra of tangles. So it is nice to
see that there is also a formulation of the 4-color theorem in terms of
the Temperley-Lieb algebra (which I won't present here).
2. Knots and physics, by Louis Kauffman, Proc. Symp. Appl. Math. 45
(1992), 131-246.
Spin networks, topology and discrete physics, by Louis Kauffman,
University of Illinois at Chicago preprint.
Vassiliev invariants and the Jones polynomial, by Louis Kauffman,
University of Illinois at Chicago preprint.
Gauss codes and quantum groups, by Louis Kauffman, University of
Illinois at Chicago preprint.
Fermions and link invariants, by Louis Kauffman
and H. Saleur, Yale University preprint YCTP-P21-91, July 5, 1991.
State models for link polynomials, by Louis Kauffman, L'Enseignement
Mathematique, 36 (1990), 1 - 37.
The Conway polynomial in R^3 and in thickened surfaces: a new
determinant formulation, by F. Jaeger, Louis Kauffman
and H. Saleur, preprint.
These are a variety of papers on knots, physics and everything.... The
more free-wheeling among you might enjoy the comments at the end of the
first paper on "knot epistemology."
I am going to a conference on gravity at UC Santa Barbara on
Friday and Saturday, which I why I am posting this early, and why I have
no time to describe the above papers. I'll talk about my usual
obsessions, and hear what other people are up to, perhaps bringing back
some words of wisdom for next week's "This Week's Finds".
-----------------------------------------------------------------------------
Previous editions of "This Week's Finds," and other expository posts
on mathematical physics, are available by anonymous ftp from
math.princeton.edu, thanks to Francis Fung. They are in the directory
/pub/fycfung/baezpapers. Please don't ask me about hep-th and gr-qc;
instead, read the sci.physics FAQ or the file preprint.info in
/pub/fycfung/baezpapers.
Archimedes Plutonium
However, I believe this is required when you see New Ohm's law as being V = CBE rather than V = CR. I have seen analogies to toll booths on highways to try to ease the discomfort of this counterintuitive. But a better explanation is that Old Physics Ohm's law V = CR is just plain wrong. When we include Resistance = Magnetic field X Electric field the parallel circuit with added resistors making overall resistance go down is now easily explained.
----Quoting from the web---
It is clear from observing the indicator bulbs in the above diagrams that the addition of more resistors causes the indicator bulb to get brighter. For parallel circuits, as the number of resistors increases, the overall current also increases. This increase in current is consistent with a decrease in overall resistance. Adding more resistors in a separate branch has the unexpected result of decreasing the overall resistance!
(snipped)
The Tollbooth Analogy
The effect of adding resistors is quite different if added in parallel compared to adding them in series. Adding more resistors in series means that there is more overall resistance; yet adding more resistors in parallel means that there is less overall resistance. The fact that one can add more resistors in parallel and produce less resistance is quite bothersome to many. An analogy may help to clarify the reason behind this initially bothersome truth.
--- end quoting from the Web ---
AP does not like counterintuitiveness in science. So whenever counterintuitive rears its ugly head, means there is probably something wrong in theory, or something missing in theory. And no analogy is going to rectify the problem, since it is a theory in error.
So it is my hunch that the "Decrease in overall resistance" is because Old Physics Ohm's law has V= CR when it really is V = CBE.
But it is late at night and will work on this tomorrow....
AP, King of Science, especially Physics
Archimedes Plutonium
AP respectfully requests Philip Hanlon Dartmouth president and Marc Tessier-Lavigne Stanford Univ president to go to their respective University student newspaper and have printed the Acknowledgement that the Slant cut in Single Cone is not an ellipse, but is in fact the Oval. The slant cut in a cylinder is indeed a ellipse, but not a single cone. For the single cone has only 1 axis of Symmetry, while the cylinder has 2 axis of Symmetry.
This is important especially for Dr. Hanlon since he taught mathematics.
So I need both of these gentlemen to Acknowledge the truth of Science. Otherwise they continue as being nothing more than trash propaganda polluters of both science and education. Acknowledge your past mistake and move forward.
3rd published book
AP's Proof-Ellipse was never a Conic Section // Math proof series, book 1 Kindle Edition
by Archimedes Plutonium (Author)
Ever since Ancient Greek Times it was thought the slant cut into a cone is the ellipse. That was false. For the slant cut in every cone is a Oval, never an Ellipse. This book is a proof that the slant cut is a oval, never the ellipse. A slant cut into the Cylinder is in fact a ellipse, but never in a cone.
Product details
• ASIN : B07PLSDQWC
• Publication date : March 11, 2019
• Language : English
• File size : 1621 KB
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•
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#11-2, 11th published book
World's First Geometry Proof of Fundamental Theorem of Calculus// Math proof series, book 2 Kindle Edition
by Archimedes Plutonium (Author)
Last revision was 15Dec2021. This is AP's 11th published book of science.
Preface:
Actually my title is too modest, for the proof that lies within this book makes it the World's First Valid Proof of Fundamental Theorem of Calculus, for in my modesty, I just wanted to emphasis that calculus was geometry and needed a geometry proof. Not being modest, there has never been a valid proof of FTC until AP's 2015 proof. This also implies that only a geometry proof of FTC constitutes a valid proof of FTC.
Calculus needs a geometry proof of Fundamental Theorem of Calculus. But none could ever be obtained in Old Math so long as they had a huge mass of mistakes, errors, fakes and con-artist trickery such as the "limit analysis". And very surprising that most math professors cannot tell the difference between a "proving something" and that of "analyzing something". As if an analysis is the same as a proof. We often analyze various things each and every day, but few if none of us consider a analysis as a proof. Yet that is what happened in the science of mathematics where they took an analysis and elevated it to the stature of being a proof, when it was never a proof.
To give a Geometry Proof of Fundamental Theorem of Calculus requires math be cleaned-up and cleaned-out of most of math's mistakes and errors. So in a sense, a Geometry FTC proof is a exercise in Consistency of all of Mathematics. In order to prove a FTC geometry proof, requires throwing out the error filled mess of Old Math. Can the Reals be the true numbers of mathematics if the Reals cannot deliver a Geometry proof of FTC? Can the functions that are not polynomial functions allow us to give a Geometry proof of FTC? Can a Coordinate System in 2D have 4 quadrants and still give a Geometry proof of FTC? Can a equation of mathematics with a number that is _not a positive decimal Grid Number_ all alone on the right side of the equation, at all times, allow us to give a Geometry proof of the FTC?
Cover Picture: Is my hand written, one page geometry proof of the Fundamental Theorem of Calculus, the world's first geometry proof of FTC, 2013-2015, by AP.
Product details
ASIN : B07PQTNHMY
Publication date : March 14, 2019
Language : English
File size : 1309 KB
Text-to-Speech : Enabled
Screen Reader : Supported
Enhanced typesetting : Enabled
X-Ray : Not Enabled
Word Wise : Not Enabled
Print length : 154 pages
Lending : Enabled
Amazon Best Sellers Rank: #128,729 Paid in Kindle Store (See Top 100 Paid in Kindle Store)
#2 in 45-Minute Science & Math Short Reads
#134 in Calculus (Books)
#20 in Calculus (Kindle Store)
y
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|/______ x
More people reading and viewing AP's newsgroup than viewing sci.math, sci.physics. So AP has decided to put all NEW WORK, to his newsgroup. And there is little wonder because in AP's newsgroups, there is only solid pure science going on, not a gang of hate spewing misfits blighting the skies.
In sci.math, sci.physics there is only stalking hate spew along with Police Drag Net Spam of no value and other than hate spew there is Police drag net spam day and night.
I re-opened the old newsgroup PAU of 1990s and there one can read my recent posts without the hassle of stalkers and spammers, Police Drag Net Spam that floods each and every day, book and solution manual spammers, off-topic-misfits, front-page-hogs, churning imbeciles, stalking mockers, suppression-bullies, and demonizers. And the taxpayer funded hate spew stalkers who ad hominem you day and night on every one of your posts.
There is no discussion of science in sci.math or sci.physics, just one long line of hate spewing stalkers followed up with Police Drag Net Spam (easy to spot-- very offtopic-- with hate charged content). And countries using sci.physics & sci.math as propaganda platforms, such as tampering in elections with their mind-rot.
Read my recent posts in peace and quiet.
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Archimedes Plutonium
Archimedes Plutonium
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Archimedes Plutonium<plutonium....@gmail.com>
Dec 31, 2023, 2:58:46 PM (4 days ago)
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I was caught doing superconductivity in my 270th book of science. And in that research saw a dire need to unify Derivative with velocity (some like to call it speed) and with Ohm's law, either the V = i *R or the New Ohm's law V = i*B*E.
This project is long overdue for physics for most think of voltage as some electrical pressure. And so we have to examine that proposition very carefully and examine what current actually means.
AP
271st book of science by AP// Derivative, velocity, New Ohm's law recalibrated to one another// physics-math by Archimedes Plutonium
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Archimedes Plutonium<plutonium....@gmail.com>
Dec 31, 2023, 3:27:24 PM (4 days ago)
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271st book of science by AP// Derivative, velocity, New Ohm's law recalibrated to one another// physics-math by Archimedes Plutonium
I decided I needed to recalibrate Derivative of calculus with velocity of physics with New Ohm's Law of electromagnetism.
Old Physics Ohms law of V= i*R and New Ohm's law of V = i*B*E.
Old Physics has the notion that Voltage is a form of pressure.
We need to make all of that absolutely clear.
We do know in calculus that Dy/Dx is the derivative and is seen as a speed or velocity, and 2nd derivative as acceleration. But we also know that V/R = i or V/(B*E) = current.
And this book aims to clarify all that obfuscation.
AP
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Archimedes Plutonium<plutonium....@gmail.com>
Dec 31, 2023, 9:05:36 PM (4 days ago)
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In Calculus math and physics the calibration is easy to make. We have calculus derivative as dy/dx and from the origin point, the amount of rise for dy divided by amount of run for dx is seen as dy is amount of distance and dx as amount of time, for velocity is distance divided by time. We simply substitute for y-axis as distance and for x axis as time and we have velocity is the derivative.
But not so easy for another division. The Voltage divided by Resistance equals current. Perhaps better is Voltage divided by (magnetic field X electric field) equals current. This is difficult for if it were easy, it would have been discovered over a 100 years ago.
And what the difficulty is, here, is that electric current is closed loop waves, not particles. In velocity and derivative we have a particle-- car say, or bicycle or runner. We do not have a particle in Voltage / (B*E) = C.
In several of my earlier works I likened the current to hoola-hoops, closed loop waves which can superposition-- that is stack all into one wave of the same size as a mere single wave is the principle of superposition.
So if we take a runner let us say that is running 10km in 2 hours would be a velocity of 5km/hour and the derivative would be 5km/hour. But that same example if a current would be 10 voltage/ 2 Resistance does not make for 5 amperage current.
What we need is to make Resistance a amount between 0 and 1 such as in probability theory values. Where 1 is superconductivity and 0 is where no current gets through.
AP
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Archimedes Plutonium<plutonium....@gmail.com>
Dec 31, 2023, 9:44:16 PM (4 days ago)
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On Sunday, December 31, 2023 at 9:05:36 PM UTC-6 Archimedes Plutonium wrote:
(snipped)
So if we take a runner let us say that is running 10km in 2 hours would be a velocity of 5km/hour and the derivative would be 5km/hour. But that same example if a current would be 10 voltage/ 2 Resistance does not make for 5 amperage current.
What we need is to make Resistance a amount between 0 and 1 such as in probability theory values. Where 1 is superconductivity and 0 is where no current gets through.
So now the formula for resistance in New Physics is (BXE) magnetic field vector cross product electric field. And vector cross product is governed by sine angles where 90 degrees equals 1 and 0 degrees equals 0. These reflect the fact that we have 2 parallel plates in a capacitor and if the plates are perfectly parallel the angle is 90 degrees with superconductivity. If the angle is 60 degrees, the plates off from being perfectly parallel by 30 degrees the sine of 30 degrees is 1/2 which means only 1/2 of the starting initial current gets through the resistor. The rest of the 1/2 current goes into the magnetic field or becomes radiated electric current as infrared heat.
Keeping in mind that electric current is not particles but is closed loop waves that superposition on one another. So if we send 10 amps into the above resistor of 30degrees sine, only 5 amps follows the circuit in output and the other 5 amps becomes part of the B field or radiated out as infrared heat.
If the angle of skew from perfect parallelism were 60 degrees then 86.6% of incoming electricity would be outgoing electricity.
In this manner we make electrical resistance be the amount of skew in the two parallel plate capacitors, from not forming parallels.
And superconduction only works when there is perfect parallelism, any skew and you have resistance.
Now capacitors have all sorts of shapes, and not just parallel plates.
AP
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Archimedes Plutonium<plutonium....@gmail.com>
Jan 3, 2024, 12:43:38 AM (yesterday)
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Alright let me review units here, for a moment.
Pressure is Force/area or kg/(meters*sec^2)
Angular momentum is kg*m^2/ sec
Voltage as given in Old Physics is kg*m^2/ (sec^3* current). I say Old Physics as they likely messed up on volt units, when it should be a form of energy kg*m^2/sec^2.
So this gives me the clue I need. That Voltage is the second derivative of Angular momentum with respect to time.
To confirm my suspicions, Linear momentum is kg*m/sec and the second derivative with respect to time is force kg*m/sec^2.
So, well, that really helps a lot for that makes logical sense. If the 2nd derivative of linear momentum is force, then the second derivative of angular momentum is Voltage.
AP, King of Science
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Archimedes Plutonium<plutonium....@gmail.com>
Jan 3, 2024, 3:23:33 PM (yesterday)
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Alright this is excellent progress, for UNITS are some of the toughest and major problems in all of science, especially physics.
It is easy to tell a stupid greenhorn of physics, especially over in sci.physics from a true physicist, for a greenhorn stumbles and falls all over units, and his/her major claim in physics is 90% that of a illusion and delusion over units.
One can say that 4 years in college of study of physics, is a study of Units, that is how important units are.
However, Units of Old Physics are in dire need of revamping and made modern, especially Voltage and the electromagnetic units. For the True Electron of Atoms is not the 0.5MeV particle but rather, the muon, stuck inside a 840MeV proton torus and doing the Faraday law with the proton. This changes all of Old Physics, even its units.
And this book is in hopes of correcting all the Units of Old Physics.
I am satisfied with yesterday's progress that the Voltage is the 2nd derivative of Angular Momentum, yielding a energy term for voltage.
As for Linear Momentum which is the first derivative of a particle in motion and then the 2nd derivative of Linear Momentum itself is a Force.
I have always been troubled with Voltage being a Pressure, ever since College teaching of it. When actually, voltage is energy.
So I am going to rely on Known facts of both Math and Physics.
In Math Geometry, we cannot go beyond Volume = L x W x H or in units as meters^3.
In Physics, we cannot go beyond Energy as E = mc^2.
Yet the Units of Old Physics routinely go beyond volume of math and energy of physics. For Old Physics was in a generations of believing in continuums and higher dimensions beyond 3rd, which created havoc upon Units.
Can we see that mc^2 is similar to L x W x H only where W and H are equal to one another?
Physics has conservation of energy. Does math need conservation laws??
Why is physics energy involve meters^2/seconds^2, yet in math we can have volumes divide in exponent 3. Volume_A^3 / Volume_B^3.
Why is calculus basically limited to the plane?? And calculus of 3rd dimension is absent???
AP, King of Science
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Archimedes Plutonium<plutonium....@gmail.com>
12:39 AM (15 hours ago)
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If I make no more progress than that of Voltage is the 2nd derivative of angular momentum which is Energy; and force is the 2nd derivative of linear momentum, will be progress well worth the effort. But I think I can solve much much more.
Voltage as given in Old Physics is kg*m^2/ (sec^3* current). I say Old Physics as they likely messed up on volt units, when it should be a form of energy kg*m^2/sec^2.
So this gives me the clue I need. That Voltage is the second derivative of Angular momentum with respect to time.
To confirm my suspicions, Linear momentum is kg*m/sec and the second derivative with respect to time is force kg*m/sec^2.
I look upon the Pressure concept as nebulous. Force/area.
This way of doing physics can allow for a huge list more of obfuscation terms and units.
Force/ temperature
Force/ angle
Force/ volume
Force/ line segment
Force/ point
Force / direction
etc etc
So if we admit Force / area, we have to admit hundreds of other silly terms and concepts.
But why is mathematics ending with exponent 3 in volume, yet physics ends in energy that is exponent 2 in Kinetic energy = 1/2 mv^2. Why does math need exponent 3 for volume units yet physics needs just exponent 2 in meters^2/seconds^2. This to me is a momentary mystery I would like to solve.
Does it have something to do with the fact that Calculus is 2 dimensional? We can only do calculus in the x-y plane. Is Calculus the cause of physics rising only to a pinnacle of exponent 2 while math rises to volume with exponent 3 in cubic-whatever?
AP, King of Science
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Archimedes Plutonium<plutonium....@gmail.com>
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On Thursday, January 4, 2024 at 12:39:21 AM UTC-6 Archimedes Plutonium wrote:
But why is mathematics ending with exponent 3 in volume, yet physics ends in energy that is exponent 2 in Kinetic energy = 1/2 mv^2. Why does math need exponent 3 for volume units yet physics needs just exponent 2 in meters^2/seconds^2. This to me is a momentary mystery I would like to solve.
Does it have something to do with the fact that Calculus is 2 dimensional? We can only do calculus in the x-y plane. Is Calculus the cause of physics rising only to a pinnacle of exponent 2 while math rises to volume with exponent 3 in cubic-whatever?
I am getting very close to solving this mystery of physics needs only exponent 2 while math needs exponent 3, a truly magnificient mystery.
I think the answer lies with an awkward thing we learned in High School. In High School we were about to learn area of triangle, having learned area of rectangle was easy. But here, area of triangle was 1/2 (base) x (height) and we had to solve for height on many occasions.
And then the reasoning came to me, we turn any triangle into a rectangle with two of them and the reason for the 1/2.
So I have New Ohm's Law as V = C*(B*E) with * being a generalized multiplication that could be scalar, vector dot product or vector cross product. Much like the formula of Volume in geometry except all is scalar there.
In physics the (BXE) is a vector cross product and represents Resistance in Old Ohm's law.
The BXE also represents the right-hand-rule, the Curl Right Hand Rule and represents the 2nd law of thermodynamics.
Now the BXE is in a plane, a planar product and looks like the same sort of operation that is done on a triangle, a generalized triangle in which we want the area. So we double the triangle to transform it into a rectangle. Take the area of rectangle and divide by 2.
So, likely the entire framework of Vector Dot Product and Vector Cross Product is a methodology of getting the area of a generalized triangle.
Because the (B*E) is a planar configuration is the reason Energy of physics needs only 2D while Geometry of math needs exponent 3 of 3D.
This also leads to the Geometry Application of moving a planar figure end up being a volume in 3D. The volume of a rectangular box is the movement as the Length of a planar cross-section of width x height.
There is the possibility here of finally consolidating what is vector dot product and vector cross product as just fancy words for finding triangle areas in a plane and then the current moving along yields the energy in physics or the volume in geometry.
We can see, in our mind's eye that a triangle is the smallest closed circuit configuration, can we not, and so the (BXE) is seeking a triangular area then multiply that over the distance of current yields energy.
AP, King of Science
Very crude dot picture of 5f6 magnetosphere of 231Pu Atom Totality
A torus shape doing the Faraday Law inside of each and every atom.
____
.-' `-.
.' ::\ ::|:: /:: `.
/ ::\::|::/:: \
; _ _ ;
| ___( O )___ |
; - - ;
\ ::/::|::\:: /
`. ::/ ::|:: \:: .'
`- _____ .-'
One of those dots in the magnetosphere is the Milky Way galaxy. And
each dot represents another galaxy. The O is the Cosmic nucleus and
certainly not as dense as what Old Physics thought because in New Physics
the interior of atoms has the Faraday law with the donut hole occupied by neutrons as storage capacitors.
I re-opened the old newsgroup PAU of 1990s and there one can read my recent posts without the hassle of spammers, govt-police drag net spam,off-topic-misfits, front-page-hogs, stalking mockers, suppression-bullies, and demonizers.
Read my recent posts in peace and quiet. If you, the reader, is wondering why AP posts this to a thread which is off topic in sci.math or sci.physics, is because some stalkers track AP, such as kibo, dan, jan who have been paid to stalk for 3 decades and when they see AP trying to post to his own thread that is on-topic they throw a impossible reCAPTcha suppression and repression at me that only wastes my time. From what AP can make out-- Google is not the only one using reCAPTcha, apparently the US govt rents out reCAPTcha. So if you see a AP post in a thread off topic, is because kooks of reCAPTcha are making it impossible for AP to post to the on-topic thread.
Read all of AP's post in peace and quiet in his newsgroup-- what sci.physics and sci.math should look like when govt spammers are not allowed in a newsgroup to wreck the newsgroup. Govt spammers have their agenda of drag net spam, and then their agenda of spy message codes, such as the "i sick, i cry" baloney, which only ends up ruining the newsgroups and why Google decided to close shop having fought govt bureaucrat mind sets for 30 years, and time to close shop.
AP kindly asks Google to let AP run all three, sci.math, sci.physics, PAU as he runs PAU, now--- all pure science, no spam and no govt b.s.
https://groups.google.com/forum/?hl=en#!forum/plutonium-atom-universe
Archimedes Plutonium
15m views-- 271st book of science by AP// Derivative, velocity, New Ohm's law recalibrated to one another// physics-math by Archimedes Plutonium