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Why is Maxwell's theory so hard to understand? (2007) [pdf]

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Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#81

Post by iorrus »

DoctorOetker wrote: post #79
iorrus wrote: post #36

Earlier quoted context omitted.

Exactly, one of the main reasons why maxwell is so hard to understand is that everything is expressed using quarternions unlike Heaviside who expressed the equations using the vector notation we see them expressed in today. In reality ‘Maxwell’s’ equations are in fact Heaviside’s.

the physics equivalent to vi/emacs flamewars doesn't need any more fuel please... https://en.wikipedia.org/wiki/Nabla_symbol#History Also curious, which of Maxwell's texts you are referencing?

“He took these books home and tried to find out. He succeeded after some trouble, but found some of the properties of vectors professedly proved were wholly incomprehensible. How could the square of a vector be negative? And Hamilton was so positive about it. After the deepest research, the youth gave it up [and] died.

My own introduction to quaternions took place in quite a different manner. Maxwell exhibited his main results in quaternionic form in his treatise. I went to Prof. Tait’s treatise to get information, and to learn how to work them. I had the same difficulties as the deceased youth, but by skipping them, was able to see that quaternions could be explored consistently in vectorial form. But on proceeding to apply quaternionics to the development of electrical theory, I found it very inconvenient. ... So I dropped out the quaternions altogether, and kept to pure scalars and vectors....” —Heaviside

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#82

Post by iorrus »

jacobolus wrote: post #75
iorrus wrote: post #36

Earlier quoted context omitted.

Exactly, one of the main reasons why maxwell is so hard to understand is that everything is expressed using quarternions unlike Heaviside who expressed the equations using the vector notation we see them expressed in today. In reality ‘Maxwell’s’ equations are in fact Heaviside’s.

Arguably the reason that generations of STEM students have been horribly confused about 3-dimensional vectors and rotations (including electric/magnetic fields), etc. is that they were reframed in the confused and non-generalizable Gibbs/Heaviside language, instead of in Grassmann/Clifford’s formalism in which vectors and bivectors can be properly described as separate types of objects. It can be so much nicer. http:…

Wow Thanks for that... fascinating.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#83

Post by srean »

ncmncm wrote: post #51
iorrus wrote: post #36

Earlier quoted context omitted.

Exactly, one of the main reasons why maxwell is so hard to understand is that everything is expressed using quarternions unlike Heaviside who expressed the equations using the vector notation we see them expressed in today. In reality ‘Maxwell’s’ equations are in fact Heaviside’s.

We should not neglect Josiah Willard Gibbs's notational contribution, either. We otherwise hear of Gibbs mainly when we mention Gibbs free energy.

Gibbs free energy and the entire body of literature on classical statistical thermodynamics. That alone would have been a enviable legacy to leave behind.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#84

Post by segfaultbuserr »

ncmncm wrote: post #51
iorrus wrote: post #36

Earlier quoted context omitted.

Exactly, one of the main reasons why maxwell is so hard to understand is that everything is expressed using quarternions unlike Heaviside who expressed the equations using the vector notation we see them expressed in today. In reality ‘Maxwell’s’ equations are in fact Heaviside’s.

We should not neglect Josiah Willard Gibbs's notational contribution, either. We otherwise hear of Gibbs mainly when we mention Gibbs free energy.

My realization about Gibbs's contributions was reading an article about Fourier analysis in electronics, the article said when you approximate a square wave by using a series of sine waves, there will be overshoots at the edges, and this problem is called "Gibbs phenomenon".

What, "Gibbs"? Wasn't he the physicist working on the physics behind chemical reactions? So he did some applied mathematics, too? It shouldn't be the same person, is it? Then I've read his Wikipedia article...

And ironically, Gibbs phenomenon was initially discovered by an unnamed mathematician Henry Wilbraham in 1848, not Gibbs, but his paper was ignored and forgotten, like many other important ones in the history of science...

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#85

Post by mikorym »

I have a copy of Maxwell's two volumes on electromagnetism and what makes it difficult to read for me is that his "equations" are actually a long list of disparate equations that describe a unified theory. It's not a simple "answer" such as E=mc^2, it is a series of such descriptor answers.

Mathematically, it is easier for me to focus on one aspect and study that aspect's equations (at a time).

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#86

Post by jules »

earthicus wrote: post #69
jules wrote: post #67

Earlier quoted context omitted.

Yes, they operate on the same objects. The main operations on differential forms are the exterior derivative d and the wedge product f /\ g. These are independent of the metric. If you add a metric you get one extra operation called the Hodge star ⋆, which is a prefix operator, so it operates as ⋆f. Alternatively, if you have a metric on your space you can define the geometric algebra derivative D and the geometric p…

Is it possible to work with geometric algebra effectively without a metric using some more complicated construction?

You can put an arbitrary metric on your space, but usually you wouldn't need metric dependent operations if you don't have a metric.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#87

Post by jules »

DoctorOetker wrote: post #76
jules wrote: post #61

Earlier quoted context omitted.

E&M is simple in the sense that the equations that govern the relationship between the charge distribution and the electromagnetic field are quite simple. And the equations governing how charges move in an electromagnetic field are quite simple. I think this is what Dyson has in mind. Understanding the implications and limitations of those equations is not simple at all. An even more blatant difficulty with E&M, rela…

with schizophrenic you mean a system of coupled differential equations? in that case I wouldn't agree with schizophrenic . or are you referring to the fact that mathematically speaking the system of differential equations is indeterministic due to mathematically perfectly valid advanced potentials? in this case I agree that there may be much more to learn from the Maxwell equations, and I have no qualms with naming t…

I mean that the theory splits into two parts: the part that tells you the fields given the charges, and the part that tells you the charges given the fields. For instance, it doesn't tell you what two charges with given initial velocities will do. (not even if you also give an initial electromagnetic field)

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#88

Post by eng101 »

adrianratnapala wrote: post #63
ncmncm wrote: post #21

It is unfortunate that Dyson neglects the role of Oliver Heaviside, again. This is in a long tradition of English neglect, ultimately traceable to Heaviside's status as a commoner. Heaviside invented the mathematical tools we still use to understand and teach Maxwell, and most of the important consequences of the theory, but Pupin, Hertz, Marconi, and deForest used his methods and took got the credit. Today Heaviside…

> This is in a long tradition of English neglect, ultimately traceable to Heaviside's status as a commoner This is unfair on the English. Glancing at their wikipedia entries, Heavyside doesn't seem any more common than [Dirac][1] or [Faraday][2]. If I had to guess at a sociological reason why Heavyside got ignored, it's that he got caught in the transition when people started learning engineering through universities…

*Heaviside

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#89

Post by Certhas »

jules wrote: post #87
DoctorOetker wrote: post #76

Earlier quoted context omitted.

with schizophrenic you mean a system of coupled differential equations? in that case I wouldn't agree with schizophrenic . or are you referring to the fact that mathematically speaking the system of differential equations is indeterministic due to mathematically perfectly valid advanced potentials? in this case I agree that there may be much more to learn from the Maxwell equations, and I have no qualms with naming t…

I mean that the theory splits into two parts: the part that tells you the fields given the charges, and the part that tells you the charges given the fields. For instance, it doesn't tell you what two charges with given initial velocities will do. (not even if you also give an initial electromagnetic field)

Well, the Lorentz force does though. And that follows from energy conservation and the Maxwell equations, e.g.:

https://physics.stackexchange.com/a/77028

So that's not really terribly deep or even true.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#90

Post by jules »

Certhas wrote: post #89
jules wrote: post #87

Earlier quoted context omitted.

I mean that the theory splits into two parts: the part that tells you the fields given the charges, and the part that tells you the charges given the fields. For instance, it doesn't tell you what two charges with given initial velocities will do. (not even if you also give an initial electromagnetic field)

Well, the Lorentz force does though. And that follows from energy conservation and the Maxwell equations, e.g.: https://physics.stackexchange.com/a/77028 So that's not really terribly deep or even true.

So what are the equations for those two charges?
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