Re:Clifford Algebra: A visual introduction (2014)
by jules on Saturday June 20, 2015 @03:44PM
(#9750841)
I do agree that GA is overhyped, but it's definitely not useless. GA brings many separate concepts into a common framework. He's a couple of things that are just ugly:
- Cross product only works in 3d
- Why is it that the cross product seems mysteriously related to determinants?
- Multivariate calculus as typically taught is very ugly
- Complex numbers for 2d, quaternions for 3d, then what?
- Rotations/reflections using linear algebra
- Cauchy's integral formula, Stokes' theorem, Helmholz decomposition, etc.
- Etcetera
GA just makes all of that much nicer and brings it together into a coherent framework.
Here's a simple problem that illustrates why GA is nice: given 3 points A,B,C in 2-d space with integer coordinates, determine whether B is on the line segment AC. Now do it in 3-d, and then in n-d.
- Cross product only works in 3d
- Why is it that the cross product seems mysteriously related to determinants?
- Multivariate calculus as typically taught is very ugly
- Complex numbers for 2d, quaternions for 3d, then what?
- Rotations/reflections using linear algebra
- Cauchy's integral formula, Stokes' theorem, Helmholz decomposition, etc.
- Etcetera
GA just makes all of that much nicer and brings it together into a coherent framework.
Here's a simple problem that illustrates why GA is nice: given 3 points A,B,C in 2-d space with integer coordinates, determine whether B is on the line segment AC. Now do it in 3-d, and then in n-d.