By standard approach I mean the typical material covered for someone studying vector calculus properly. This will be stuff like differential forms and the basics of tensors, manifolds and multilinear maps at the undergrad level. Differential geometry and cohomology are examples of courses which build on them.
I agree with you that pseudovectors, cross products and vector calculus are a terribly adhoc way to teach this stuff but a course covering linear algebra with differential forms elegantly unifies, corrects and generalizes them. Standard is also in contrast to the geometric algebra/calculus alternate path.
If you can’t invert vectors, you aren’t studying vector calculus properly. ;-)
Differential forms are a half-baked formalism.
Unfortunately I don’t know of any great undergraduate level geometric calculus textbooks. Ideally there would be something like Hubbard & Hubbard’s book (http://matrixeditions.com/5thUnifiedApproach.html) written using GA as a formalism.
I agree with you that pseudovectors, cross products and vector calculus are a terribly adhoc way to teach this stuff but a course covering linear algebra with differential forms elegantly unifies, corrects and generalizes them. Standard is also in contrast to the geometric algebra/calculus alternate path.