This is a great paper.
As a counterpoint, one place where determinants are incredibly useful is in Hartree-Fock theory, where they effective encode the Pauli exclusion principle (or anti-symmetry requirements) of atomic orbitals.
As a counterpoint, one place where determinants are incredibly useful is in Hartree-Fock theory, where they effective encode the Pauli exclusion principle (or anti-symmetry requirements) of atomic orbitals.
(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).)
John Hamal Hubbard, Barbara Burke Hubbard - Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach
> https://en.wikipedia.org/wiki/Seven-dimensional_cross_produc...
EDIT: Nore precisely: A common way to axiomatize the cross product yields a cross products exactly in dimension 3 and 7.