21–30 of 38 posts
Nobody ever writes Maxell's equations using "components", the title sounds like a straw man argument. The rest of the text is well written, but hopelessly useless without a comparison with the typical way to write Maxwell's equations using differential forms (which turns out to be essentially identical to geometric algebra).
Nobody ever writes Maxell's equations using "components", the title sounds like a straw man argument. The rest of the text is well written, but hopelessly useless without a comparison with the typical way to write Maxwell's equations using differential forms (which turns out to be essentially identical to geometric algebra).
If I understand correctly, Maxwell wrote Maxwell's equations using components. That's why they took 20 equations.
Earlier quoted context omitted.
You're mistaken, GA does have a Hodge star, as I've explained many times before https://grassmann.crucialflow.com/dev/algebra The exterior product can be derived from the geometric product, so differential forms occur in geometric algebra.
You can easily define it, that's what I meant saying that you can do the same things in practice, but it's not usually defined (at least in the books and articles I've read), and certainly it is not so ubiquitous as in DFs texts. And, of course, the exterior product is contained in the geometric product. I guess that, in the same way, you could define a geometric product operator when using a DFs formulation. Would y…
Recently 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.
(2x2 real matrices are also isomorphic to the geometric algebra of the 2-dimensional pseudo-Euclidean vector plane with signature (+, -), under a different interpretation.)
By comparison the quaternions are the even subalgebra of the geometric algebra of 3-dimensional Euclidean vector space with signature (+,+,+), consisting of only the 1 scalar and 3 bivector components. Or under a different interpretation are isomorphic to the full geometric algebra of the 2-dimensional vector plane with signature (-, -). They can be represented as Pauli matrices.
For more on this see the papers and books of Garret Sobczyk, https://garretstar.com/secciones/publications/publications.h... ; for example the recent https://www.garretstar.com/sobczyk09-mar-2020.pdf
If a point is the intersection of two lines, and a line is the intersection of two planes, then what structures intersect to give a plane? Two 3D spaces? And so the intersection of 2 4D spaces should give a 3D space? How do I express these ideas with GA?
Total non expert here but your fascinating question got me thinking out loud. Two 3D volumes in a 3D space obviously intersect to give another 3D volume, unless they're tangent. So at first glance I'd be tempted to say no, but... Two overlapping polygons in a 2D space also define another 2D polygon. Two overlapping line segments in a 1D space define another line segment. You only get a reduced-dimension object at the…
Nobody ever writes Maxell's equations using "components", the title sounds like a straw man argument. The rest of the text is well written, but hopelessly useless without a comparison with the typical way to write Maxwell's equations using differential forms (which turns out to be essentially identical to geometric algebra).
Nobody ever writes Maxell's equations using "components", the title sounds like a straw man argument. The rest of the text is well written, but hopelessly useless without a comparison with the typical way to write Maxwell's equations using differential forms (which turns out to be essentially identical to geometric algebra).
Earlier quoted context omitted.
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 don…
If I remember correctly, the Hodge star much more closely belongs to geometric algebras than it does to exterior algebras, since you need a nondegenerate bilinear form to define the Hodge star, from which you can just as easily define the geometric product, and from the geometric product the Hodge star. To make an analogy, it sounds a bit like you're asking whether inner product spaces and vector spaces are equivalen…
Earlier quoted context omitted.
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 don…
If I remember correctly, the Hodge star much more closely belongs to geometric algebras than it does to exterior algebras, since you need a nondegenerate bilinear form to define the Hodge star, from which you can just as easily define the geometric product, and from the geometric product the Hodge star. To make an analogy, it sounds a bit like you're asking whether inner product spaces and vector spaces are equivalen…
I work with people who use Euler angles to express rotations, and it's a horrible world. I learned quaternions in my day, and there are some obvious advantages. When I discovered GA some years ago, it was really eye opening. It makes quaternions an easy to explain concept, even intuitive, and I've used it since then, not only in my own work but also to teach other people. Then, I learned about differential forms, and it's also very interesting, I think I could base my "intuitive explanations" in this other paradigm, but I'm not sure I should. I do not think it makes a big difference in my particular case, but as I said I find this "competition" a bit frustrating, and am trying to understand it better.
I cannot discuss with a mathematician if the Hodge star is a GA or a DFs concept, but I have found it all over the place when reading about DFs, and not so in GA related material (though I have a vague idea about how the Hodge star operator can easily be defined in GAs using the pseudoscalar). But is this really my choice? I have listened opinions about which one is more general, but not really convincing arguments (at least not arguments that are obvious to me).
Thanks for your explanation. I think I need to have a deeper look at this stuff. I like to get lost in these rabbit holes, but sometimes it goes a bit over my head.
Earlier quoted context omitted.
You can easily define it, that's what I meant saying that you can do the same things in practice, but it's not usually defined (at least in the books and articles I've read), and certainly it is not so ubiquitous as in DFs texts. And, of course, the exterior product is contained in the geometric product. I guess that, in the same way, you could define a geometric product operator when using a DFs formulation. Would y…
No, I would say differential forms occur in geometric algebra, not the other way around.
You obviously know more than me about this, so I will ask you a slightly different question: if I learn GA well enough and totally ignore differential forms, what will I miss?