4 ms·
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 e
by experiment0 9y ago
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.
https://en.wikipedia.org/wiki/Hartree–Fock_method https://en.wikipedia.org/wiki/Hartree–Fock_method
- nategri 9y agoAlso: Cross products.
- marcelluspye 9y agoBased on this article, I would venture to guess that Axler is also not a fan of the cross product, though, so he wouldn't feel that much is lost. Cross products are practically a vector-calculus hack that lack generality and risk obscuring intuition at the mathematical level.
- ajkjk 9y agoI'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).)
- zodiac 9y agoWhich books would you recommend to learn this wedge product / differential forms approach to linear algebra and complex numbers?
- selimthegrim 9y agoDoesn't Spivak get into them?
- wolfgke 9y agoMichael Spivak has written multiple books. Which one do you mean?
- tgb 9y agoCalculus on Manifolds. I'd recommend Hubbard and Hubbard over that as it's a little easier read with the same material.
- wolfgke 9y ago> I'd recommend Hubbard and Hubbard over that as it's a little easier read with the same material. John Hamal Hubbard, Barbara Burke Hubbard - Vector Calculus, Linear Algebra and Differential Forms: A Unified Approach
- Koshkin 9y agoOne problem with cross product, it only exists in three dimensions.
- wolfgke 9y agoNo, it exists in dimension 3 and 7: > https://en.wikipedia.org/wiki/Seven-dimensional_cross_product 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.
- sixo 9y agoA wedge product exists in all dimensions but only in 3 & 7 is it identifiable with a unique vector.