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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.

Qmechanic
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Doron Shafrir
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  • In terms of E&M, the 4 vector calculus Maxwell's equations contract to 2 when using the language of tensor calculus (2-tensors and 4-gradients) and differential forms (exterior derivatives and Hodge duals). Remarkably, Maxwell's equations can be expressed using one equation when working in a Clifford algebra. So yes, E&M has a "cross-product free approach". Here is a paper that introduces E&M more generally using differential forms: http://www.jpier.org/PIER/pier148/09.14063009.pdf – JG123 Dec 21 '19 at 21:44
  • Vectors and cross-products can be systematically replaced by differential forms. A classical source is Arnold, Mathematical Methods of Classical Mechanics. At a more elementary level, Bressoud, Second Year Calculus: From Celestial Mechanics To Special Relativity. – Conifold Dec 22 '19 at 04:44