I worded that in an unclear way. My main beef is that GA is just way clunkier than differential forms, which are clearly the "right" approach if you want to approach the subject from a theoretical perspective. I see no advantage over the usual treatment, and many disadvantages.
I'm a skeptic too, but I might not be the intended user for the GA formalism. Please explain your reasons.
My skepticism of the supposedly superior pedagogy of Geometric Algebra is the following:
- 3D vector algebra with the cross product operation and the dot product operation is fairly easy and intuitive. Its replacement by GA might not be so easy. So maybe GA should be introduced after the vector formalism.
- An arbitrary element of a Geometric Algebra might not have a geometric meaning. For instance, some elements of a GA are vectors, while some are scalars, but there are also these exotic mixed quantities which are scalars plus vectors. This is pretty hard for me to understand intuitively.
- An arbitrary matrix has a geometric meaning. It's essentially just a linear transformation. By contrast, I don't feel that an arbitrary element of a geometric algebra has a geometric meaning.
- Consider those elements of a Geometric Algebra which represent rotations -- they are called rotors. Observe that if "z" is a rotor then the element "-z" is also a rotor which stands for the same rotation as "z". So there is more to a rotor than whatever rotation it describes. This seems very unintuitive and advanced. (I know that this behaviour has applications for the study of spin-1/2 particles in quantum physics).
I also have trouble understanding where the rule for multiplying two elements of a geometric algebra comes from. It's an operation, introduced from seemingly nowhere, which happens to have some applications in some areas. But I'm not comfortable with a multiplication rule being introduced out of nowhere without being derived out of something. The claim that it has a consistent geometric meaning from which it can be derived is never justified. My criticisms are therefore largely pedagogical.
I think your reasons are good and essentially what I would give. I also think differential forms are a much theoretically cleaner way to express the same concepts.
Further, teaching differential forms prepares my students to engage with the (vast majority of the) existing math and physics literature. Teaching geometric algebra doesn't.
The practical reasons boil down to: I have to teach the standard stuff because otherwise they can't read the literature. Having done that, what's the marginal benefit of teaching GA? Not a lot.
On the other hand, maybe GA can help with differential forms. Differential forms involves exterior algebra, and I feel like some aspects of exterior algebra are elaborated upon in an insightful way by GA. For instance, the grade-2 elements of an Exterior Algebra can be understood as angular velocities in many circumstances. In GA, this is captured by the exponential map that sends grade-2 elements to rotors. I don't know if this can be helpful for teaching purposes.
