Differential geometry, geometric algebra etc

November 28, 2014 — September 19, 2023

🏗 A pragmatic guide to manifold wrangling.

Figure 1

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

See information geometry.

🏗

1 Differential forms formalism

2 Clifford Algebra formalism

Hestenes’ school.

  • Cavendish laboratory’s introduction

  • 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

  • Hestene’s most infamous bit of rhetoric

  • stack exchange arguments.

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

Absil, Mahony, and Sepulchre. 2008. Optimization algorithms on matrix manifolds.
Amari, Shun’Ichi. 1990. Differential-Geometrical Methods in Statistics.
Amari, Shunʼichi. 2001. Information Geometry on Hierarchy of Probability Distributions.” IEEE Transactions on Information Theory.
Bachman. 2003. A Geometric Approach to Differential Forms.
Bowen, and Wang. 1976. Introduction to Vectors and Tensors, Vol 1: Linear and Multilinear Algebra.
Bowen, and Wang. 2006. Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis.
Brandstetter, Berg, Welling, et al. 2022. Clifford Neural Layers for PDE Modeling.” In.
Brody, and Rivier. 1995. Geometrical Aspects of Statistical Mechanics.” Phys. Rev. E.
Chisolm. 2012. Geometric Algebra.” arXiv:1205.5935 [Math-Ph].
Do Carmo. 1992. Riemannian Geometry.
Donnelly, and Rogers. 2005. Symplectic Integrators: An Introduction.” American Journal of Physics.
Doran, and Lasenby. 2003. Geometric Algebra for Physicists.
Dorst, Fontijne, and Mann. 2010. Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry.
Dullemond, and Peeters. 2021. Introduction to Tensor Calculus.”
Edwards. 1994. Advanced Calculus: A Differential Forms Approach.
Feragen, and Hauberg. 2016. Open Problem: Kernel Methods on Manifolds and Metric Spaces. What Is the Probability of a Positive Definite Geodesic Exponential Kernel? In Conference on Learning Theory.
Flanders. 1989. Differential Forms with Applications to the Physical Sciences.
Frankel. 2011. The Geometry of Physics: An Introduction.
Hestenes, David. 1988. Universal Geometric Algebra.”
———. 1991. The Design of Linear Algebra and Geometry.” Acta Applicandae Mathematica.
Hestenes, Prof David. 2001. Old Wine in New Bottles: A New Algebraic Framework for Computational Geometry.” In Geometric Algebra with Applications in Science and Engineering.
Hestenes, David. 2003. Spacetime Physics with Geometric Algebra.” American Journal of Physics.
Hestenes, David, and Sobczyk. 1984. Clifford Algebra to Geometric Calculus a Unified Language for Mathematics and Physics.
Isham. 1989. Modern Differential Geometry for Physicists.
Lasenby, Lasenby, and Doran. 2000. A Unified Mathematical Language for Physics and Engineering in the 21st Century.” Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences.
Lauritzen. 1987. “Statistical Manifolds.” In Differential Geometry in Statistical Inference.
Lee. 1972. Differential and physical geometry.
Leibon, Pauls, Rockmore, et al. 2010. “Statistical Learning for Complex Systems an Example-Driven Introduction.”
Macdonald. 2016. A Survey of Geometric Algebra and Geometric Calculus.” Advances in Applied Clifford Algebras.
Moore. 1993. “Braids in Classical Dynamics.” Physical Review Letters.
Needham. 2021. Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts.
Nielsen. 2018. An Elementary Introduction to Information Geometry.” arXiv:1808.08271 [Cs, Math, Stat].
Porteous. 1995. Clifford Algebras and the Classical Groups. Cambridge Studies in Advanced Mathematics 50.
Ruhe, Gupta, de Keninck, et al. 2023. Geometric Clifford Algebra Networks.” In arXiv Preprint arXiv:2302.06594.
Spivak. 1998. Calculus on manifolds: a modern approach to classical theorems of advanced calculus. Mathematics monograph series.
Tarantola. 2006. Elements for Physics: Quantities, Qualities, and Intrinsic Theories.
Warnick, K.F., and Arnold. 1996. Electromagnetic Green Functions Using Differential Forms.” Journal of Electromagnetic Waves and Applications.
Warnick, K.F., Arnold, and Selfridge. 1996. Differential Forms in Electromagnetic Field Theory.” In Antennas and Propagation Society International Symposium, 1996. AP-S. Digest.
Warnick, Karl F., Selfridge, and Arnold. 1995. Electromagnetic Boundary Conditions and Differential Forms.” IEE Proceedings-Microwaves, Antennas and Propagation.
———. 1996. Electromagnetics Made Easy: Differential Forms as a Teaching Tool.” In 2013 IEEE Frontiers in Education Conference (FIE).
Warnick, Karl F., Selfridge, and Arnold. 1997. Teaching Electromagnetic Field Theory Using Differential Forms.” IEEE Transactions on Education.
Weintraub. 2014. Differential Forms, Second Edition: Theory and Practice.