🏗 A pragmatic guide to manifold wrangling.
Key words: Clifford algebra, differential forms… Bollu’s Handy list of differential geometry definitions.
See information geometry.
Differential forms formalism
- Flanders, Differential forms (Flanders 1989) is a wildly popular classic available for cheap used
- Frankel, the geometry of physics (Frankel 2011) attempts to update Flanders, trading slightly improved pedagogy for slightly more physics specificity.
- Bachman’s book (Bachman 2003) seems popular and is free.
- John Baez’s high speed introduction
Clifford Algebra formalism
The connection between physics teaching and research at its deepest level can be illuminated by Physics Education Research (PER). For students and scientists alike, what they know and learn about physics is profoundly shaped by the conceptual tools at their command. Physicists employ a miscellaneous assortment of mathematical tools in ways that contribute to a fragmentation of knowledge. We can do better! Research on the design and use of mathematical systems provides a guide for designing a unified mathematical language for the whole of physics that facilitates learning and enhances physical insight. This has produced a comprehensive language called Geometric Algebra, which I introduce with emphasis on how it simplifies and integrates classical and quantum physics. Introducing research-based reform into a conservative physics curriculum is a challenge for the emerging PER community. Join the fun!
My purpose in this chapter is to introduce you to a powerful new algebraic model for Euclidean space with all sorts of applications to computer-aided geometry, robotics, computer vision and the like. A detailed description and analysis of the model is soon to be published elsewhere , so I can concentrate on highlights here, although with a slightly different formulation that I find more convenient for applications. Also, I can assume that this audience is familiar with Geometric Algebra, so we can proceed rapidly without belaboring the basics
Alan Bromborsky, An introduction to Geometric Algebra and Calculus.
Alan Macdonald, Geometric Algebra:
Geometric algebra and its extension to geometric calculus unify, simplify, and generalize vast areas of mathematics that involve geometric ideas, including linear algebra, multivariable calculus, real analysis, complex analysis, and euclidean, noneuclidean, and projective geometry. They provide a unified mathematical language for physics (classical and quantum mechanics, electrodynamics, relativity), the geometrical aspects of computer science (e.g., graphics, robotics, computer vision), and engineering.