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.
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?
To make an analogy, it sounds a bit like you're asking whether inner product spaces and vector spaces are equivalent. Every geometric algebra gives rise to a Hodge star, and an exterior algebra, and so on, but exterior algebras are a much more general concept, so they're less powerful until you tack that extra structure on.
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.
The exterior product can be derived from the geometric product, so differential forms occur in geometric algebra.
In any case, you did not attempt to answer my original question. Are GA and DFs just different ways to define "equivalent" concepts or is there some more fundamental difference that I am missing?
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?
Differentials are a concept that the comes from doing calculus on manifolds, and exterior products of differentials are just used for tracking information about oriented volumes.
To answer your question (switching differentials forms for for exterior algebras), you wont miss anything, as the wedge product is part of a GA.
ABSTRACT: In this paper, we explicate the suggested benefits of Clifford’s geometric algebra (GA) when applied to the field of electrical engineering. Engineers are always interested in keeping formulas as simple or compact as possible, and we illustrate that geometric algebra does provide such a simplified representation in many cases. We also demonstrate an additional structural check provided by GA for formulas in addition to the usual checking of physical dimensions. Naturally, there is an initial learning curve when applying a new method, but it appears to be worth the effort, as we show significantly simplified formulas, greater intuition, and improved problem solving in many cases.
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.
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
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).
Join the discord https://discord.gg/vGY6pPk.
Check out a demo https://observablehq.com/@enkimute/animated-orbits
Similarly in an ambient 3-d space, 2 generic planes intersect to give a line. But the planes can also be coincident or parallel. And in higher ambient spaces can intersect at only a point, rather than a line, or even fail to intersect in a non-parallel way.
In 4-d space, the intersection of two generic 3-d spaces does indeed give a plane, with exactly similar caveats.
The standard GA doesn't directly represent general lines or planes, however. The elements are the equivalent of "vectors" rather than "points", and always go through 0. The obvious way to handle these are parameterizing the lines and surfaces, but you're essentially working with equations for the surfaces, and keeping track of the variables.
The slick way of handling it is with _projective_ geometric algebra, and intersections turn into "meets". The meet of two parallel lines (planes) is now a "point (line) at infinity", and of a (line, plane) with itself is the line (plane) again. Skew lines have a meet of 0 (not the point 0, the number 0).
GA has been well discussed here, so, the other half:
Projectivization is a fairly standard trick even for normal geometry. It's adding an additional dimension, which is in most contexts just set to 1. (Projective actually just means treating all points on a ray as equivalent; this loses the dimension you just gained
It lets rotations and translations be treated in a nearly uniform manner, and lets you do rotations around points that aren't the origin. It's used all over the place in much graphics code (usually under the name homogeneous coördinates). The last section of https://en.wikipedia.org/wiki/Homogeneous_coordinates discusses this briefly.
For online resources for the combination: another commenter has recommended https://bivector.net/ , and it looks okay, with sections specifically on 2 and 3 dimensional projective geometric algebra.
There is also the nice C++ header library klein: https://www.jeremyong.com/klein/
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 intersection if they're intersecting in a higher dimensional space. Otherwise they can't intersect "at an angle".
So I'd say purely by following the pattern that it must be that two 3D spaces intersecting "at an angle" within a 4D space define a plane by their intersection.
The source of confusion here is that we're mixing up (bounded) volumetric shapes, line segments, and planar polygons, with their infinite counterparts - the lines, points, planes, and infinite volumes. An infinite object has no shape - it is infinite in some dimensions and zero in all others - this is what gives us those clear well-defined intersection objects, which are either zero or infinite. A bounded object has a shape - it has a boundary that is not infinite. When you intersect such objects within their shared subspaces (two line segments on the same line, two polygons on the same plane, two volumes in the same space) you get either a null, or an object of the same type. This is obvious when you make one of the objects infinite - an intersection of a line segment with its line gives you the same line segment. So the reason you are having difficulty imagining two somethings that will intersect in a plane is that a plane is an infinite object, so only two infinite objects can intersect in a plane. So you need two infinite volumes, that pass through the same 4D space, but are not the same volume. These volumes share a two-dimensional subspace - the area of 4D space they intersect in is a plane. It's infinite in two dimensions, and zero-sized in two others.
Thanks for the more thorough explanation!