Re:Maxwell's Equations first published 150 years ago
by bainsfather on Tuesday November 24, 2015 @05:26AM
(#10620044)
I've read that book, it is very good. A fantastic story/subject, though probably not for a general audience, just those interested in a detailed history of Heaviside and his work. It's nice to see a well-written 'technical' history book - so often such books are written by someone who doesn't really understand the subject - this is an exception.
You are mistaken about Laplace Transforms - Heaviside used a different method which was rather ad-hoc - sometimes it works, sometimes not. For the same problems, we now use Laplace Transforms instead, because they are better/cleaner.
Re:Maxwell's Equations first published 150 years ago
by juped on Tuesday November 24, 2015 @12:26AM
(#10619268)
Heaviside was somewhat written out of history because he was considered a crank during his lifetime. His reputation is basically rehabilitated these days, but it's only physicists/mathematicians/engineers who have heard of him.
Re:Maxwell's Equations first published 150 years ago
by femto on Monday November 23, 2015 @09:30PM
(#10618816)
From the Wikipedia article:
"From 1882 to 1902, except for three years, he contributed regular articles to the trade paper The Electrician, which wished to improve its standing, for which he was paid £40 per year. This was hardly enough to live on, but his demands were very small and he was doing what he most wanted to. "
Re:Maxwell's Equations first published 150 years ago
by dboreham on Monday November 23, 2015 @09:11PM
(#10618735)
In reading up on this the other day, when that Guardian article appeared, I was intrigued to find that there is a conspiracy theory types who advance the idea that Heaviside was the bad guy because he removed important parts of Maxwell's work that explains, well...X-Files type stuff. Read on: http://www.cheniere.org/references/maxwell.htm
Re:Maxwell's Equations first published 150 years ago
by mhartl on Tuesday November 24, 2015 @02:14AM
(#10619543)
Some HN readers might be interested to know that Maxwell's equations can be written in a single expression (with one auxiliary condition).
The first step is to define the Faraday tensorF, which can be written as a 4x4 antisymmetric matrix. Such a matrix has 16 elements, but the diagonal is all zeroes (by antisymmetry: x = -x => x = 0), and the 6 off-diagonal elements on one side of the diagonal are simply the negatives of their mirrored values. This leaves 16 - 4 - 6 = 6 independent elements—exactly enough to contain the 6 elements of the electric and magnetic field vectors (3 each for E and B), which in fact are exactly the elements of the Faraday tensor.
In terms of F, the Maxwell equations for the divergence of B and the curl of E can be combined in the single equation
dF = 0
where d is the exterior derivative. As with ordinary potential theory, where a vector field with zero curl can be written as the gradient of a scalar function, a tensor whose exterior derivative is zero can be derived from another tensor, called in this case the 4-potential A (which contains both the ordinary electric potential and the magnetic vector potential):
F = dA (1)
(This is the auxiliary condition alluded to above.) The second expression for F, corresponding to the divergence of E and the curl of B, combines the exterior derivative and the Hodge star operator ⭑ [1]:
d⭑F = J (2)
where J is the 4-current density (which contains both the charge density and the 3-dimensional current density) [2]. Substituting (1) for F in (2) then yields Maxwell's equations in a single expression:
d⭑dA = J Maxwell's equation(s)
This equation is perhaps a more compact and elegant choice for those T-shirts: "God said: d⭑dA = J—and there was light!"
[1]: The ⭑ character may not display in some browsers. In this case, you can use an asterisk instead.
[2]: Eq. (2) is written in "God's units", equivalent to SI with μ₀ = ε₀ = 1 => c = 1.
Re:Maxwell's Equations first published 150 years ago
by McKayDavis on Monday November 23, 2015 @08:31PM
(#10618596)
...and in two days it is the 100th anniversary of Einstein's publication of The Field Equations of Gravitation. (The original paper on General Relativity).
Re:Maxwell's Equations first published 150 years ago
by cnvogel on Tuesday November 24, 2015 @02:03AM
(#10619517)
Of course, like every article published by a newspaper or magazine, its only purpose is to carry links to other articles published by this newspaper, and omit any reference to the original source.
Re:Maxwell's Equations first published 150 years ago
by 4ad on Tuesday November 24, 2015 @06:43AM
(#10620228)
The vector calculus formulation is not cumbersome and confusing, and we should not replace it. Many problems are simplest in vector calculus. Also, vector calculus formulation gives some insights that are not evident in other formulations. It's also extremely enlightening from a pedagogic perspective when you are learning physics. Of course if you're a mathematician and you don't want to learn physics, than vector calculus is just a distraction because it's idiosyncratic and doesn't connect with anything else.
The differential forms formulation, which in many other cases (but not all!) is more useful to work with than the vector calculus formulation should probably be replaced with geometric algebra though. They are basically the same thing, but differential forms are more idiosyncratic, and some symmetries are not apparent. Neither are useful as an introduction to the aspiring physicist or engineer though.
Re:Maxwell's Equations first published 150 years ago
by iheartmemcache on Tuesday November 24, 2015 @02:45AM
(#10619615)
Yep. I'm real glad Hestenes has been championing VA/VC so hard. That first text is a perfect supplementary reading that any undergraduate should read. Who was the author?
I skipped the a few and went straight to the last PDF (the Survey) and coming from a stronger foundational math background, I felt right at home. (I'd imagine those who went down the physics route would be thrown off by the notation though, so I understand the author's preface). He mentions physics and CS applications, but I really think this has structural engineering applications as well -- e.g. R^4 seems like a perfect domain to explore bidirectional tensile and shear strength of materials (especially composite materials/aggregates), then perform some FEA with all of the simple operations within that come inherent to operating within the bounds of the 4-vector form. I mean that's a real trivial example (and I obviously don't know enough about structural engineering to make a good analogy) but it seems like one of those cases where you move the problem set lifted into another domain, perform your manipulation then finally lower back into your original (co-)domain. (Frequently done in algebraic topology, most famously by Perelman on Poincare)
Re:Maxwell's Equations first published 150 years ago
by semi-extrinsic on Tuesday November 24, 2015 @04:22AM
(#10619877)
I think there is a very good reason why structural engineers never see this type of thing, and it's because most structural engineers don't like math. They like clicking buttons in their CAD/FEA/BIM/<insert-latest-TLA> computer system.
The people who write those systems on the other hand enjoy this math very much, and tend to use Galerkin as a verb and will casually point out just how wonderful the Lax-Milgram theorem is.
Supposedly they're only known as "Maxwell Equations" now because Einstein began his 1905 paper naming them such. Prior to that they were known by other names. Before even my time though.
Frank Yang's talk (and the associated paper, but that's only available behind a paywall) might be of interest to those looking for more background: https://www.youtube.com/watch?v=SG343kojbnU
... Who took the original 20 equations and simplified them to the 4 we normally see.
That's a fairly major footnote, on an interesting career. He seems to be one of those people who quietly had a significant impact on the future.
Good book: http://www.amazon.com/Oliver-Heaviside-Electrical-Genius-Vic...
You are mistaken about Laplace Transforms - Heaviside used a different method which was rather ad-hoc - sometimes it works, sometimes not. For the same problems, we now use Laplace Transforms instead, because they are better/cleaner.
"From 1882 to 1902, except for three years, he contributed regular articles to the trade paper The Electrician, which wished to improve its standing, for which he was paid £40 per year. This was hardly enough to live on, but his demands were very small and he was doing what he most wanted to. "
The prototypical hacker?
The first one to reduce them to a single equation, in relativistic form.
Be nice to have some citations.
The first step is to define the Faraday tensor F, which can be written as a 4x4 antisymmetric matrix. Such a matrix has 16 elements, but the diagonal is all zeroes (by antisymmetry: x = -x => x = 0), and the 6 off-diagonal elements on one side of the diagonal are simply the negatives of their mirrored values. This leaves 16 - 4 - 6 = 6 independent elements—exactly enough to contain the 6 elements of the electric and magnetic field vectors (3 each for E and B), which in fact are exactly the elements of the Faraday tensor.
In terms of F, the Maxwell equations for the divergence of B and the curl of E can be combined in the single equation
where d is the exterior derivative. As with ordinary potential theory, where a vector field with zero curl can be written as the gradient of a scalar function, a tensor whose exterior derivative is zero can be derived from another tensor, called in this case the 4-potential A (which contains both the ordinary electric potential and the magnetic vector potential): (This is the auxiliary condition alluded to above.) The second expression for F, corresponding to the divergence of E and the curl of B, combines the exterior derivative and the Hodge star operator ⭑ [1]: where J is the 4-current density (which contains both the charge density and the 3-dimensional current density) [2]. Substituting (1) for F in (2) then yields Maxwell's equations in a single expression: This equation is perhaps a more compact and elegant choice for those T-shirts: "God said: d⭑dA = J—and there was light!"[1]: The ⭑ character may not display in some browsers. In this case, you can use an asterisk instead.
[2]: Eq. (2) is written in "God's units", equivalent to SI with μ₀ = ε₀ = 1 => c = 1.
[1] http://einsteinpapers.press.princeton.edu/vol6-trans/129
[2] https://en.wikipedia.org/wiki/History_of_general_relativity#...
[3] http://www-history.mcs.st-and.ac.uk/HistTopics/General_relat...
So here's the original publication: http://rstl.royalsocietypublishing.org/content/155.toc
Philosophical Transactions of the Royal Society of London, For the year ⅯⅮⅭⅭⅭⅬⅩⅤ, Vol. 155
http://rstl.royalsocietypublishing.org/content/155/local/fro... (Title page and Contents) http://rstl.royalsocietypublishing.org/content/155/459.full.... (Article)
Gibbs/Heaviside vectors are cumbersome and confusing, and the faster we can dump them as a society the better.
The differential forms formulation, which in many other cases (but not all!) is more useful to work with than the vector calculus formulation should probably be replaced with geometric algebra though. They are basically the same thing, but differential forms are more idiosyncratic, and some symmetries are not apparent. Neither are useful as an introduction to the aspiring physicist or engineer though.
Or open any of these and search for “Maxwell”: http://www.av8n.com/physics/maxwell-ga.htm http://arxiv.org/pdf/1101.3619.pdf http://ieeexplore.ieee.org/ielx7/5/6879517/06876131.pdf http://geocalc.clas.asu.edu/pdf/SpacetimePhysics.pdf http://faculty.luther.edu/~macdonal/GA&GC.pdf
I skipped the a few and went straight to the last PDF (the Survey) and coming from a stronger foundational math background, I felt right at home. (I'd imagine those who went down the physics route would be thrown off by the notation though, so I understand the author's preface). He mentions physics and CS applications, but I really think this has structural engineering applications as well -- e.g. R^4 seems like a perfect domain to explore bidirectional tensile and shear strength of materials (especially composite materials/aggregates), then perform some FEA with all of the simple operations within that come inherent to operating within the bounds of the 4-vector form. I mean that's a real trivial example (and I obviously don't know enough about structural engineering to make a good analogy) but it seems like one of those cases where you move the problem set lifted into another domain, perform your manipulation then finally lower back into your original (co-)domain. (Frequently done in algebraic topology, most famously by Perelman on Poincare)
The people who write those systems on the other hand enjoy this math very much, and tend to use Galerkin as a verb and will casually point out just how wonderful the Lax-Milgram theorem is.
http://www.setileague.org/articles/ham/maxwell.pdf
Supposedly they're only known as "Maxwell Equations" now because Einstein began his 1905 paper naming them such. Prior to that they were known by other names. Before even my time though.
Frank Yang's talk (and the associated paper, but that's only available behind a paywall) might be of interest to those looking for more background: https://www.youtube.com/watch?v=SG343kojbnU
https://en.wikipedia.org/wiki/Vector_Analysis