I did degrees in both math and engineering, please give me the benefit of the doubt that I'm not making a 1st year university level mistake. In the specific ex

I did degrees in both math and engineering, please give me the benefit of the doubt that I'm not making a 1st year university level mistake. In the specific examples you give, 2-forms could be introduced instead to yield the final formula for the Jacobian. I don't think this is an overkill, because it's possible to use 2-forms without precisely proving the theorems etc about them. This is more convenient for math/physics people, and engineers care more about the final formula than about how exactly it was derived. As another comment here pointed to, the idea that linear algebra is better off without determinants was popularized by Axler - I'm not making this opinion up out of nowhere.