It's just some of the more advanced theory you'd get from studying modules repackaged a bit.
Basically the extra stuff that is usually skipped in first year linear algebra courses are the symmetric and asymmetric (often called exterior) products. These form algebras, of course. The exterior product, or wedge product, has a natural interpretation in terms of signed areas (or volumes) and from this you get the determinant as a volume form.
These are the natural generalizations of dot products (inner products) and wedge products (exterior products).
You can take a vector and associate it with a 1 form (asymmetric algebra or exterior algebra), and then multiply two vectors to get a 2 form using the standard wedge product, etc. In dimension 3, the space of 2 forms is dual to the space of 1 forms and so you can "multiply" two vectors to get a third vector. That is all that's going on here.
Actually a good multi-variable calculus class will cover most of this stuff as you need some motivation for Jacobian volume forms used to calculate areas and volumes under change of basis, and dot/wedge products are useful for generalizations of the Gauss divergence theorem and the generalized fundamental theorem that says the integral over a function, f, on the n-1 dimensional boundary of a shape is the differential of the integral of the shape.
Moreover any class on Riemannian geometry will give you all the linear algebra you need as well.
One thing I would caution students with is that by using somewhat non-standard jargon they may not understand how to generalize this stuff to n-dimensions, nor will the connections between, say, determinants and wedge-forms be clear, or dot products and angles be fully understood if only the n=3 cases is emphasized. Only in n=3 can you multiply two vectors to get a vector. But fun fact: in dimension 3k you can multiply two k-forms to get a third k-form (as the space of 2k forms is dual to the space of k forms in n=3k). If you think there is this new thing called "geometric algebra" other than usual tensor products, it may not be obvious how things generalize to n != 3.
> But no, there is nothing new here beyond marketing.
Yes, that's my sense too. Of course cross products, wedge products etc make sense and that's just standard mathematics, but the part that I haven't really seen the point of is to form the algebra where all these forms live side by side.
It doesn't seem like a useful "fusing", in the way that say the complex plane is.
Of course it's very cool that sub-algebras in 2 or 3-space in GA are isomorphic to the complex plane or even quaternions, but it still feels a bit made up.
For a concrete example, one youtuber showed how Maxwells equations simplified to a single equation if you introduce an operator that is a combination of div and curl, and also a new kind of physical entity that combines the electrical and magnetic fields.
This is of course cool, but what I want to know is if this new operator makes some physical sense, and if the new multi dimensional field has any physical meaning. If they don't, it just seems like a parlour trick.
Not saying they actually don't, but I haven't seen any deeper explanations of it.
Not the same concrete example, but one where I do find the Geometric Algebra version substantially more insightful, is the treatment of rigid body mechanics in the geometric algebra of the Euclidean group (R_{n,0,1}).
It has the dual quaternions as even subalgebra (in 3D), and unifies all linear and angular aspects. It leads to remarkable new insights, as removing the need for force-couples (pure angular acceleration is caused by pushing along a line at infinity), while pure linear acceleration is caused by forces along lines through the center of mass.
These geometric ideas are independent of dimension - forces, both angular and linear are always lines. The treatment of inertia becomes a duality map, and things like Steiners theorem are not needed at all.
On top of this, the separation of the metric that sets GA apart means that this formulation of rigid body dynamics works not only in flat Euclidean space, but unmodified in the Spherical and Hyperbolic geometries. (by a simple change of metric of the projective dimension).
Well, I think the point is that in rigid body dynamics, the configuration and phase spaces naturally form a manifold and then the equations of motion are in terms of differential forms on the cotangent bundle of the these manifolds. This is commonly expressed in terms of the language of exterior algebras, hodge duals, etc. That's what is driving all of this, and is usually covered in a good class on mathematical physics. Again, there is nothing new here except marketing, but marketing plays an important and useful role.
I remember for a long time, people coming from the math end of things would look down a bit on physicists laboriously working everything out in complex tensor notation when there are these elegant canonical descriptions arising from differential geometry that look very simple and beautiful and are completely coordinate-invariant.
But then when you want to actually calculate something, you end up doing all the painful tensor contractions anyway, so the physicists would likewise often lookdown on the mathematicians for writing these simple one liners that described all of mechanics but not really understanding how to calculate stuff.
So if repackaging some of the basic facts of differential geometry as "Geometric Algebra" gets physicists to be excited about it, then that's a good thing. Just like repackaging some of the laborious tensor calculus computations into differential geometry has gotten a lot of mathematicians excited about physics. It really is much more pleasant to work in a coordinate-free manner using differential structures associated to the natural manifold suggested by the problem, rather than being stuck in euclidean space and needing to deal with lots of fictional forces and complex change of basis formulas.
> For a concrete example, one youtuber showed how Maxwells equations simplified to a single equation if you introduce an operator that is a combination of div and curl, and also a new kind of physical entity that combines the electrical and magnetic fields.
Back when I was in university, we covered this in our differential geometry class. And yes, you'd use more abstract concepts like curvature, hodge dual, and exterior product.
Maxwells equations in any dimension can be reduced to:
dF = 0 and d*F = 0
That's two equations, not one, but you can introduce a new D = (d, d*) and then get DF=0 if you want.
The advantage here is the d, and F have all the old physical meanings. F is curvature, which is the electro-magnetic field E+B, and d is the derivative (exterior derivative, but that is the derivative needed in calculus).
That looks pretty much equivalent, and they even have a version where F = E + B, just like in the youtube video. But my question is if this F, or the tensor version for that matter, has any physical meaning?
I think the point is that when learning EM, it's a lot about developing intuition by visualizing in your head how the E and B fields behave in different situations, and how charges interact with them etc. You know, a lot of holding your right hand in the "physicist handshake" position and twisting it.
Sure, the same information is contained in the F tensor, but it doesn't have a similar straightforward geometric intuition.
It’s pretty easy to think about a normal three dimensional vector field representing the magnetic field, the trouble for me is when you combine the two into one tensor field.
Looking at the link reminds me that this entity is used outside of GA, but it still feels a bit weird. More weird than say complex currents, but maybe it actually isn’t.
n=3 concern applies ewually well to every formulation, as it's a peculiarity if human scale physics that's commonly used as an application of linear algebra and anaylsis (visualizing linear transformations and rotations, point-line-plane geometry in 3D space, curl, Maxwell's equations)
Basically the extra stuff that is usually skipped in first year linear algebra courses are the symmetric and asymmetric (often called exterior) products. These form algebras, of course. The exterior product, or wedge product, has a natural interpretation in terms of signed areas (or volumes) and from this you get the determinant as a volume form.
These are the natural generalizations of dot products (inner products) and wedge products (exterior products).
You can take a vector and associate it with a 1 form (asymmetric algebra or exterior algebra), and then multiply two vectors to get a 2 form using the standard wedge product, etc. In dimension 3, the space of 2 forms is dual to the space of 1 forms and so you can "multiply" two vectors to get a third vector. That is all that's going on here.
Actually a good multi-variable calculus class will cover most of this stuff as you need some motivation for Jacobian volume forms used to calculate areas and volumes under change of basis, and dot/wedge products are useful for generalizations of the Gauss divergence theorem and the generalized fundamental theorem that says the integral over a function, f, on the n-1 dimensional boundary of a shape is the differential of the integral of the shape.
Moreover any class on Riemannian geometry will give you all the linear algebra you need as well.
One thing I would caution students with is that by using somewhat non-standard jargon they may not understand how to generalize this stuff to n-dimensions, nor will the connections between, say, determinants and wedge-forms be clear, or dot products and angles be fully understood if only the n=3 cases is emphasized. Only in n=3 can you multiply two vectors to get a vector. But fun fact: in dimension 3k you can multiply two k-forms to get a third k-form (as the space of 2k forms is dual to the space of k forms in n=3k). If you think there is this new thing called "geometric algebra" other than usual tensor products, it may not be obvious how things generalize to n != 3.