Electromagnetism using geometric algebra versus components
1–10 of 38 posts
Re: Electromagnetism using geometric algebra versus components
#2Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the vector?
With GA, you associate the magnetic field with an oriented plane element. Oriented planes have this flip-mirroring property. Because space is 3D, you have exactly one vector to which this plane element is perpendicular, so you can identify the field with a vector. This is called the Hodge duality, and this particular form is accidental to 3D space.
If you need more convincing, the magnitude of the magnetic field is given by the cross-product, which calculates the area spanned by two vectors.
You could argue that you don't need GA for this, only to use the exterior product instead of the cross product. But by using an algebra that contains both vectors and oriented areas, you can effectively sum the electric and magnetic fields. This will give you pretty compact and beautiful equations, which is the essence of this article.
Re: Electromagnetism using geometric algebra versus components
#3Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the v…
For anyone looking for a sane way to study this stuff, what finally made sense to me was (1) study discrete differential geometry from Keenan Crane's notes: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ ; (2) Read these notes on geometric algebra that formally lay down what the hell a GA space is and what operations one can perform on them: https://arxiv.org/pdf/1205.5935.pdf (Geometric Algebra: Eric Chisolm)
Re: Electromagnetism using geometric algebra versus components
#4Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the v…
For anyone looking for a sane way to study this stuff, what finally made sense to me was (1) study discrete differential geometry from Keenan Crane's notes: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ ; (2) Read these notes on geometric algebra that formally lay down what the hell a GA space is and what operations one can perform on them: https://arxiv.org/pdf/1205.5935.pdf (Geometric Algebra: Eric Chisolm)
For clicking convenience: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ (the link above is broken since HN includes the ]; in the URL).
Re: Electromagnetism using geometric algebra versus components
#5Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the v…
For anyone looking for a sane way to study this stuff, what finally made sense to me was (1) study discrete differential geometry from Keenan Crane's notes: https://www.cs.cmu.edu/~kmcrane/Projects/DDG/ ; (2) Read these notes on geometric algebra that formally lay down what the hell a GA space is and what operations one can perform on them: https://arxiv.org/pdf/1205.5935.pdf (Geometric Algebra: Eric Chisolm)
This is also an excellent and interactive blog-post building up to the understanding of how rotors can replace quaternions in GA: https://marctenbosch.com/quaternions/
Re: Electromagnetism using geometric algebra versus components
#6Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the v…
You still don’t need geometric algebra to combine the magnetic and electric fields, if you view them as differential forms on 4-dimensional spacetime. From what I can tell, this approach is basically equivalent to what this article does. You could directly translate everything into the language of differential forms, because the only geometric products here are in fact just exterior products.
Re: Electromagnetism using geometric algebra versus components
#7You should check out the bivector community https://bivector.net/ dedicated to Geometric algebra.
Join the discord https://discord.gg/vGY6pPk.
Check out a demo https://observablehq.com/@enkimute/animated-orbits
Re: Electromagnetism using geometric algebra versus components
#8Recently I made a video presentation about geometric algebra, including the formulation of Maxwell's equations in GA:
https://youtu.be/7hlDRLEhc8o
Also, my implementation of geometric algebra in the Julia language, Grassmann.jl https://github.com/chakravala/Grassmann.jl
It helped me truly understand Maxwell's equations for the first time, understanding that it is not just some physical artifact but actually a natural foundational idea in pure mathematics applicable to physics.
Re: Electromagnetism using geometric algebra versus components
#9Personally, an appealing part of GA is clarifying that the magnetic field is best viewed as an "oriented area", instead of a pseudo-vector [1], which is a concept that is frequently ill-defined. Usually, lecturers say that the magnetic field is essentially a vector, but to be careful that it flips directions when you mirror space. But how then can you check this by just looking at the usual three coordinates of the v…
You still don’t need geometric algebra to combine the magnetic and electric fields, if you view them as differential forms on 4-dimensional spacetime. From what I can tell, this approach is basically equivalent to what this article does. You could directly translate everything into the language of differential forms, because the only geometric products here are in fact just exterior products.
This "competition" between geometric algebra and differential forms makes me uncomfortable.
As far as I see (and I'm not an expert), they are just different ways to express very similar concepts. They are still not exactly the same, since the geometric product is not defined in DFs, and there is no hodge star operator in GA, for example, but everything you can do using one formalism in practice can also be easily done using the other one. What am I missing?
Re: Electromagnetism using geometric algebra versus components
#10Recently I made a video presentation about geometric algebra, including the formulation of Maxwell's equations in GA: https://youtu.be/7hlDRLEhc8o Also, my implementation of geometric algebra in the Julia language, Grassmann.jl https://github.com/chakravala/Grassmann.jl It helped me truly understand Maxwell's equations for the first time, understanding that it is not just some physical artifact but actually a natural…
That’s cool.
What do you make of the algebra of the dihedrons? https://youtu.be/lqH4BLHGsFw . It’s a “sister algebra” of the quaternions.