# Differential geometry, geometric algebra etc

November 28, 2014 — September 19, 2023

🏗 A pragmatic guide to manifold wrangling.

Key words: Clifford algebra, differential forms… Bollu’s Handy list of differential geometry definitions.

See information geometry.

🏗

## 1 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

## 2 Clifford Algebra formalism

Hestenes’ school.

Reforming the matehmatical language of physics

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!

Old Wine in New Bottles: A new algebraic framework for computational geometry

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 [1], 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

Handy texts:

Tristan Needham, Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts (Needham 2021).

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.

## 3 Tooling

computer algebra system Cadabra supports Exterior derivatives.

## 4 References

*Optimization algorithms on matrix manifolds*.

*Differential-Geometrical Methods in Statistics*.

*IEEE Transactions on Information Theory*.

*A Geometric Approach to Differential Forms*.

*Introduction to Vectors and Tensors, Vol 1: Linear and Multilinear Algebra*.

*Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis*.

*Phys. Rev. E*.

*arXiv:1205.5935 [Math-Ph]*.

*Riemannian Geometry*.

*American Journal of Physics*.

*Geometric Algebra for Physicists*.

*Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry*.

*Advanced Calculus: A Differential Forms Approach*.

*Conference on Learning Theory*.

*Differential Forms with Applications to the Physical Sciences*.

*The Geometry of Physics: An Introduction*.

*Acta Applicandae Mathematica*.

*Geometric Algebra with Applications in Science and Engineering*.

*American Journal of Physics*.

*Clifford Algebra to Geometric Calculus a Unified Language for Mathematics and Physics*.

*Modern Differential Geometry for Physicists*.

*Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences*.

*Differential Geometry in Statistical Inference*.

*Differential and physical geometry*.

*Advances in Applied Clifford Algebras*.

*Physical Review Letters*.

*Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts*.

*arXiv:1808.08271 [Cs, Math, Stat]*.

*Clifford Algebras and the Classical Groups*. Cambridge Studies in Advanced Mathematics 50.

*arXiv Preprint arXiv:2302.06594*.

*Calculus on manifolds: a modern approach to classical theorems of advanced calculus*. Mathematics monograph series.

*Elements for Physics: Quantities, Qualities, and Intrinsic Theories*.

*Journal of Electromagnetic Waves and Applications*.

*Antennas and Propagation Society International Symposium, 1996. AP-S. Digest*.

*IEE Proceedings-Microwaves, Antennas and Propagation*.

*2013 IEEE Frontiers in Education Conference (FIE)*.

*IEEE Transactions on Education*.

*Differential Forms, Second Edition: Theory and Practice*.