4 ms·

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 equa

by iorrus 8y ago

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.

ncmncm 8y ago
We should not neglect Josiah Willard Gibbs's notational contribution, either. We otherwise hear of Gibbs mainly when we mention Gibbs free energy.
- srean 8y ago
  Gibbs free energy and the entire body of literature on classical statistical thermodynamics. That alone would have been a enviable legacy to leave behind.
- segfaultbuserr 8y ago
  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...
jacobolus 8y ago
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://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
- iorrus 8y ago
  Wow Thanks for that... fascinating.
- MaxBarraclough 8y ago
  Reminds me, I've been meaning to read this [0] blog post for a while now. (See also the HN discussion [1].) My clueless intuition tells me its points may be analogous to your linked document. (It mentions Heaviside, at least.) [0] https://www.gamedev.net/articles/programming/math-and-physics/do-we-really-need-quaternions-r1199/ https://www.gamedev.net/articles/programming/math-and-physic... [1] https://news.ycombinator.com/item?id=18365433 https://news.ycombinator.com/item?id=18365433
  - jacobolus 8y ago
    Perhaps start with https://www.shapeoperator.com/2016/12/12/sunset-geometry/ https://www.shapeoperator.com/2016/12/12/sunset-geometry/ for a concrete example.
    - MaxBarraclough 8y ago
      Looks good, thanks.
- whatshisface 8y ago
  Geometric algebra obscures the point that what really matters is the algebraic structure. There are many ways to construct things that behave the same way (subgroups of matrices, vectors and the cross product, and so on), and while it is nice to invent one construction that gives you everything, you risk loosing sight of the fact that each construction is arbitrary and what really matters is its algebraic structure. That's why it's good to expose students to many partial, fragmented devices, so that they will realize the deeper point that underlies them all. With spin matrices, it is obvious that they are arbitrary manifestations of a group, but if a student were to spend their entire education manipulating blades they might start getting the idea that in some sense the universe was "made out of them."
DoctorOetker 8y ago
the physics equivalent to vi/emacs flamewars doesn't need any more fuel please... https://en.wikipedia.org/wiki/Nabla_symbol#History https://en.wikipedia.org/wiki/Nabla_symbol#History Also curious, which of Maxwell's texts you are referencing?
- iorrus 8y ago
  “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
wnoise 8y ago
It was my understanding that the original Maxwell equations used separate letters for each component rather than indices or any modern method of writing down the components as a unified thing (so the four expand to at least 12 separate equations (20, once you factor in constitutive and continuity equations)). Later on, he did use quaternions to organize like quantities, as we would use vectors, but still generally worked component-by-component. But yes, the most common modern versions are effectively due to Heaviside.

by iorrus 8y ago

ncmncm 8y ago

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

srean 8y ago
Gibbs free energy and the entire body of literature on classical statistical thermodynamics. That alone would have been a enviable legacy to leave behind.
segfaultbuserr 8y ago
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...

jacobolus 8y ago

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://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf

iorrus 8y ago
Wow Thanks for that... fascinating.
MaxBarraclough 8y ago
Reminds me, I've been meaning to read this [0] blog post for a while now. (See also the HN discussion [1].) My clueless intuition tells me its points may be analogous to your linked document. (It mentions Heaviside, at least.) [0] https://www.gamedev.net/articles/programming/math-and-physics/do-we-really-need-quaternions-r1199/ https://www.gamedev.net/articles/programming/math-and-physic... [1] https://news.ycombinator.com/item?id=18365433 https://news.ycombinator.com/item?id=18365433
- jacobolus 8y ago
  Perhaps start with https://www.shapeoperator.com/2016/12/12/sunset-geometry/ https://www.shapeoperator.com/2016/12/12/sunset-geometry/ for a concrete example.
  - MaxBarraclough 8y ago
    Looks good, thanks.
whatshisface 8y ago
Geometric algebra obscures the point that what really matters is the algebraic structure. There are many ways to construct things that behave the same way (subgroups of matrices, vectors and the cross product, and so on), and while it is nice to invent one construction that gives you everything, you risk loosing sight of the fact that each construction is arbitrary and what really matters is its algebraic structure. That's why it's good to expose students to many partial, fragmented devices, so that they will realize the deeper point that underlies them all. With spin matrices, it is obvious that they are arbitrary manifestations of a group, but if a student were to spend their entire education manipulating blades they might start getting the idea that in some sense the universe was "made out of them."

DoctorOetker 8y ago

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

iorrus 8y ago
“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

wnoise 8y ago

It was my understanding that the original Maxwell equations used separate letters for each component rather than indices or any modern method of writing down the components as a unified thing (so the four expand to at least 12 separate equations (20, once you factor in constitutive and continuity equations)). Later on, he did use quaternions to organize like quantities, as we would use vectors, but still generally worked component-by-component. But yes, the most common modern versions are effectively due to Heaviside.