🏗 A pragmatic guide to manifold wrangling, whatever your formalism.
Key words: Clifford algebra, differential forms… Bollu’s Handy list of differential geometry definitions.
See information geometry.
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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 focus.
- Bachman’s book (Bachman 2003) seems popular and is free.
- John Baez’s high speed introduction
Clifford Algebra formalism
Hestenes’ school.
- Cavendish laboratory’s introduction
- Hestene’s most famous bit of rhetoric
- 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.
References
Absil, P.-A, R Mahony, and R Sepulchre. 2008. Optimization algorithms on matrix manifolds. Princeton, N.J.; Woodstock: Princeton University Press.
Amari, Shun’Ichi. 1990. Differential-Geometrical Methods in Statistics. Springer.
Amari, Shunʼichi. 2001. “Information Geometry on Hierarchy of Probability Distributions.” IEEE Transactions on Information Theory 47: 1701–11.
Bachman, David. 2003. A Geometric Approach to Differential Forms.
Bowen, Ray M., and C. C. Wang. 1976. Introduction to Vectors and Tensors, Vol 1: Linear and Multilinear Algebra. Plenum Press.
Bowen, Ray M., and C.-C. Wang. 2006. Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis.
Brody, Dorje, and Nicolas Rivier. 1995. “Geometrical Aspects of Statistical Mechanics.” Phys. Rev. E 51 (2): 1006–11.
Chisolm, Eric. 2012. “Geometric Algebra.” arXiv:1205.5935 [Math-Ph], May.
Do Carmo, Manfredo P. 1992. Riemannian Geometry. Birkhauser.
Donnelly, Denis, and Edwin Rogers. 2005. “Symplectic Integrators: An Introduction.” American Journal of Physics 73 (10): 938–45.
Doran, Chris, and A. N. Lasenby. 2003. Geometric Algebra for Physicists. Cambridge ; New York: Cambridge University Press.
Dorst, Leo, Daniel Fontijne, and Stephen Mann. 2010. Geometric Algebra for Computer Science: An Object-Oriented Approach to Geometry. Morgan Kaufmann.
Dullemond, Kees, and Kasper Peeters. 2021. “Introduction to Tensor Calculus.”
Edwards, Harold M. 1994. Advanced Calculus: A Differential Forms Approach. Birkhauser.
Feragen, Aasa, and Søren 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, 1647–50. PMLR.
Flanders, Harley. 1989. Differential Forms with Applications to the Physical Sciences. Dover Publications.
Frankel, Theodore. 2011. The Geometry of Physics: An Introduction. 3 edition. Cambridge ; New York: Cambridge University Press.
Hestenes, David. 1988. “Universal Geometric Algebra.”
———. 1991. “The Design of Linear Algebra and Geometry.” Acta Applicandae Mathematica 23 (1): 65–93.
———. 2003. “Spacetime Physics with Geometric Algebra.” American Journal of Physics 71 (7): 691–714.
Hestenes, David, and Garret Sobczyk. 1984. Clifford Algebra to Geometric Calculus a Unified Language for Mathematics and Physics. Dordrecht: Springer Netherlands.
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, edited by Dr Eduardo Bayro Corrochano and Prof Garret Sobczyk, 3–17. Birkhäuser Boston.
Isham, Chris J. 1989. Modern Differential Geometry for Physicists. Allied Publishers.
Lasenby, Joan, Anthony N. Lasenby, and Chris J. L. 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 358 (1765): 21–39.
Lauritzen, S L. 1987. “Statistical Manifolds.” In Differential Geometry in Statistical Inference, 164. JSTOR.
Lee, Jeffrey Marc. 1972. Differential and physical geometry.
Leibon, Gregory, Scott Pauls, Dan Rockmore, and Robert Savell. 2010. “Statistical Learning for Complex Systems an Example-Driven Introduction.”
Macdonald, Alan. 2016. “A Survey of Geometric Algebra and Geometric Calculus.” Advances in Applied Clifford Algebras, April, 1–39.
Moore, Cristopher. 1993. “Braids in Classical Dynamics.” Physical Review Letters 70 (24): 3675–79.
Needham, Tristan. 2021. Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts. Princeton, NJ: Princeton University Press.
Nielsen, Frank. 2018. “An Elementary Introduction to Information Geometry.” arXiv:1808.08271 [Cs, Math, Stat], August.
Porteous, Ian R. 1995. Clifford Algebras and the Classical Groups. Cambridge Studies in Advanced Mathematics 50. Cambridge ; New York: Cambridge University Press.
Spivak, Michael. 1998. Calculus on manifolds: a modern approach to classical theorems of advanced calculus. 27. printing. Mathematics monograph series. Cambridge, Mass: Perseus Books.
Tarantola, Albert. 2006. Elements for Physics: Quantities, Qualities, and Intrinsic Theories. Berlin ; New York: Springer.
Warnick, Karl F., Richard H. Selfridge, and David V. Arnold. 1995. “Electromagnetic Boundary Conditions and Differential Forms.” IEE Proceedings-Microwaves, Antennas and Propagation 142 (4): 326–32.
———. 1996. “Electromagnetics Made Easy: Differential Forms as a Teaching Tool.” In 2013 IEEE Frontiers in Education Conference (FIE), 3:1508–12. IEEE.
Warnick, Karl F., Richard H. Selfridge, and D.V. Arnold. 1997. “Teaching Electromagnetic Field Theory Using Differential Forms.” IEEE Transactions on Education 40 (1): 53–68.
Warnick, K.F., and D.V. Arnold. 1996. “Electromagnetic Green Functions Using Differential Forms.” Journal of Electromagnetic Waves and Applications 10 (3): 427–38.
Warnick, K.F., D.V. Arnold, and R.H. Selfridge. 1996. “Differential Forms in Electromagnetic Field Theory.” In Antennas and Propagation Society International Symposium, 1996. AP-S. Digest, 2:1474–1477 vol.2.
Weintraub, Steven H. 2014. Differential Forms, Second Edition: Theory and Practice. Academic Press.
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