I'm sure it's out there somewhere, but it would be an appropriate corollary to the OP if there was a "Down with Cross Products", which argues that multivectors

I'm sure it's out there somewhere, but it would be an appropriate corollary to the OP if there was a "Down with Cross Products", which argues that multivectors and wedge products should be taught instead of cross products in multivariable calculus. Then, determinants are "the wedge product of N linearly independent vectors", and cross products are "the wedge product of 2 vectors in 3 dimensions", which gives a bivector and trivially encodes their pseudovector properties. (Also, surface normals in integrals are bivectors, the 'i' of complex analysis is the bivector resulting from wedge product x^y, and e^(i theta) is the exponential map applied to the i operator, and (del wedge vector-function f) is the (bivector-valued) curl while (del wedge bivector-function g) is the (scalar valued) divergence (and that's why del(del(f)) = 0).) (But differential forms should probably be omitted in a first course, because they get hairy quickly and are hard to wrap one's head around. It's enough to know that dxdy in integrals is actually dx^dy, and therefore the Jacobian appears when changing variables because of the factor that appears from dx'^dy' = dx'(x,y)^dy'(x,y).)