You can write down Maxwell’s equations using the geometric algebra easily, and it will make them better! They will be obviously coordinate independent.
Pretty much anything with a cross product will be better written using geometric algebra.
One of their main uses is to prove the generalized Stokes theorem in n dimensions.
3D rotors are isomorphic to quaternions, so go ahead and rename your variables :).
So, to be as opinionated as the g'parent comment:
The cross product is a hack which only works in certain dimensionalities, whereas the wedge product is the underlying idea, which works in all circumstances.
To be less opinionated:
The cross product is inconvenient because what it gives you aren't the "usual" vectors, they're axial vectors, which behave differently under mirror reflection than all of your other vectors do.
Whether you call it the cross product or not is just semantics, but it does exist in terms of the exterior product.
It's difficult to find online, but it's constructed directly in Spivak. A quote:
"It is uncommon in mathematics to have a "product" that depends on more than two factors. In the case of two vectors v,w in R^3, we obtain a more conventional looking product, v X w in R^3. For this reason it is sometimes maintained that the cross product. can be defined only on R^3" - Calculus on Manifolds, pg.84
Most graduate students read this (or did five years ago).
I never said the wedge was the only way to generalize the cross product, I just said the cross product itself wasn't general.
We used to think as negative numbers being a generalization of the integers, so that's some food for thought. I'm sure once quantum mechanics dominates solving eigenvalue problems will be a high-school level problem, so we'll end up having complex numbers losing their complexity. We don't call them "real" numbers outside of math circles anymore.
To say that the cross product itself is a hack is a bit of a stretch though, it can easily be generalized and I think it's quite natural.
> We used to think as negative numbers being a generalization of the integers
"used to" ?
> We don't call them "real" numbers outside of math circles anymore.
Sure we do ?
To clarify, we used to think of integers as just the natural numbers. Integer was a colloquial word meaning "whole, entire", so there was presumably a discussion about how negative numbers were whole or entire, though I vaguely remember this historical story. My point is just that at some point negative numbers were seen as a advanced extension of the natural numbers. Even zero was seen as an unnatural extension, which is why there is a confusion to this date as to whether "natural" numbers include zero.
ref: http://mathforum.org/library/drmath/view/57212.html
Not the best reference, but I've read it elsewhere, so maybe it's a myth, but someone can trace it to the source if they want.
> We don't call them "real" numbers outside of math circles anymore.
I mean laypeople don't know what the real numbers are in distinction to the rationals, for example.
Then again, with these conversations, the N < Z < Q < R < C... classification automatically pops in your mind if you did mathematics in last classes of high school (that's going to be a LOT of laypeople ! Of course, a lot of them, those that rarely encounter them, might then forget about this classification.)
And integers are still called "natural integers", and negative numbers are NOT called "natural" ?