Cross products are used everywhere in classical mechanics: torque, angular velocity, angular momentum, the Coriolis force, rigid body mechanics. This is even more so in classical electromagnetism.
Coming from a mathematical background, I'm very uncomfortable with the cross product being ubiquitous for several reasons: It limits the dimension to 3 where obviously no such limitation is necessary, and some identities involving the cross and dot products are highly confusing to me.
Has anybody tried to give an account of classical mechanics & electromagnetism banning the use of cross products alltogether?
Such an approach can vary from being quite elementary, replacing "pseudo-vectors" with anti-symmetric 3X3 matrices, or more sophisticated, using the language of differential geometry: wedge products, differential forms etc.
