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Geometric algebra is a bit awkward notationally. Physicist prefer to use an alternative notation based on exterior calculus that does provide a compact represen

by pa7x1 6y ago

Geometric algebra is a bit awkward notationally. Physicist prefer to use an alternative notation based on exterior calculus that does provide a compact representation of Maxwell equations: dF=J dF=0 Geometric algebra produces two equations too, by the way, not one.

OscarCunningham 6y ago
You can even write them in terms of the four-vector A rather than F (related by F = dA) to reduce them to ddA = J.
- danharaj 6y ago
  escape your hodge duals fam
  - OscarCunningham 6y ago
    Ha that's funny. I'd been copying my notation from the parent post, and I assumed 'd' was just their notation. I didn't realise they meant to say *d* (apparently HN doesn't let you escape asterisks).
- tobinfricke 6y ago
  Discovering that the Maxwell equations can be written in this succinct form is kind of mind-blowing, but I'm wondering whether it provides any additional insight? It seems like one needs to do a significant "unpacking" to actually use the equation or gain insight from it. I would love to hear an explanation of electrodynamics starting with "star ddA = J".
  - pa7x1 6y ago
    You get new insights, that F is a curvature 2-form. Written in this way it's explicit that EM is also a geometric theory. This observation opens the door to Yang-Mills theories which are behind all the Standard Model.
  - jabl 6y ago
    Yes, I kinda agree. The underlying physics is, by definition, the same, and what you gain by reducing the number of equations you lose by having more complicated "objects" and needing more advanced maths to handle them (e.g. the electromagnetic field tensor, external calculus and whatnot vs. just vector fields and basic vector calculus). To get an intuition of the physics, I think the traditional 4 equation form is actually more useful, as you can construct toy examples and study the equations one at a time in isolation. Where the more advanced formulations are useful, and actually are used, is for stuff like relativistic physics where 4-vectors, curved spacetime etc. are needed and not just a gimmick. But for more down-to-earth applications of electrodynamics like antennas, field propagation in various forms of matter etc., the classical version is fine.
adamnemecek 6y ago
It's one equation in geometric algebra. https://www.av8n.com/physics/maxwell-ga.htm#eq-max-ga https://www.av8n.com/physics/maxwell-ga.htm#eq-max-ga Also there is a new community for geometric algbera people https://bivector.net/ https://bivector.net/. Check out the demo https://observablehq.com/@enkimute/animated-orbits https://observablehq.com/@enkimute/animated-orbits Watch the SIGGRAPH talk https://youtu.be/tX4H_ctggYo https://youtu.be/tX4H_ctggYo Join the discord https://discord.gg/vGY6pPk https://discord.gg/vGY6pPk

by pa7x1 6y ago

OscarCunningham 6y ago

You can even write them in terms of the four-vector A rather than F (related by F = dA) to reduce them to ddA = J.

danharaj 6y ago
escape your hodge duals fam
- OscarCunningham 6y ago
  Ha that's funny. I'd been copying my notation from the parent post, and I assumed 'd' was just their notation. I didn't realise they meant to say *d* (apparently HN doesn't let you escape asterisks).
tobinfricke 6y ago
Discovering that the Maxwell equations can be written in this succinct form is kind of mind-blowing, but I'm wondering whether it provides any additional insight? It seems like one needs to do a significant "unpacking" to actually use the equation or gain insight from it. I would love to hear an explanation of electrodynamics starting with "star ddA = J".
- pa7x1 6y ago
  You get new insights, that F is a curvature 2-form. Written in this way it's explicit that EM is also a geometric theory. This observation opens the door to Yang-Mills theories which are behind all the Standard Model.
- jabl 6y ago
  Yes, I kinda agree. The underlying physics is, by definition, the same, and what you gain by reducing the number of equations you lose by having more complicated "objects" and needing more advanced maths to handle them (e.g. the electromagnetic field tensor, external calculus and whatnot vs. just vector fields and basic vector calculus). To get an intuition of the physics, I think the traditional 4 equation form is actually more useful, as you can construct toy examples and study the equations one at a time in isolation. Where the more advanced formulations are useful, and actually are used, is for stuff like relativistic physics where 4-vectors, curved spacetime etc. are needed and not just a gimmick. But for more down-to-earth applications of electrodynamics like antennas, field propagation in various forms of matter etc., the classical version is fine.

adamnemecek 6y ago

It's one equation in geometric algebra. https://www.av8n.com/physics/maxwell-ga.htm#eq-max-ga https://www.av8n.com/physics/maxwell-ga.htm#eq-max-ga Also there is a new community for geometric algbera people https://bivector.net/ https://bivector.net/. Check out the demo https://observablehq.com/@enkimute/animated-orbits https://observablehq.com/@enkimute/animated-orbits Watch the SIGGRAPH talk https://youtu.be/tX4H_ctggYo https://youtu.be/tX4H_ctggYo Join the discord https://discord.gg/vGY6pPk https://discord.gg/vGY6pPk