Universal Geometric Calculus

See also Rheology, Vectors and Values, Grassmann, Lawvere , Mathematical Economics and Capital Theory.

Must combine with Tensor Analysis.

“Universal” should include fullest Clifford Algebras (not only “Geometric” over reals) and related combinatorial “calculus” like:

Remarks on Invariant geometric calculus. Cayley-Grassmann algebras

and geometric Clifford algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 129

P Bravi, A. Brini

and

Grassmann geometric calculus, invariant theory and superalgebras . . . .. 151

A. Brini, F. Regonati, A. G. B. Teolis

both in:

ACCS

@book{book:945516,

title = {Algebraic Combinatorics and Computer Science: A Tribute to Gian-Carlo Rota},

author = {H. Crapo (auth.), H. Crapo, D. Senato (eds.)},

publisher = {Springer-Verlag Mailand},

isbn = {978-88-470-2159-4,978-88-470-2107-5},

year = {2001},

series = {},

edition = {1},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=51161aff15b090a4aa459c0e172df0da}

}

Would also like to understand via Category Theory and include essence of VA “Variational Analysis” (Rockafellar and Wets) and RV “Random Variation” (Patrick Muldowney) for path based integrals used in both stochastic finance with Loeb spaces and in Feynman QM (General Integral better conceptually than Lebesgue and usual measure theory).

But first need to understand NFM (and probably lots of Category Theory and Algebra before able to tackle any of above).

Hopefully will get there by parallel study of first Lawvere and Schaunel’s recommended initial follow ons from CM “Conceptual Mathematics” together with Walter Noll on FDS “Finite Dimensional Spaces” and perhaps Steve Awodey CT “Category Theory”, Freyd and Scedrov CA “Categories, Allegories”, Paolo Aluffi AC0 “Algebra Chapter 0”.

Hopefully avoid getting distracted or bogged down by Taylor PFM “Practical Foundations of Mathematics” or HTT “Homotopy Type Theory”.

The series from CM seems to include enough Topos theory and Synthetic Differential Geometry to have a useful perspective so better start there before attempting anything beyond LAGA and VAGC below.

NFM

@book{book:961857,
title = {New Foundations in Mathematics: The Geometric Concept of Number},
author = {Garret Sobczyk (auth.)},
publisher = {Birkhäuser Basel},
isbn = {978-0-8176-8384-9,978-0-8176-8385-6},
year = {2013},
series = {},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=6128FDB4F5CB6F933B7E456810F0A104}}

Start with (junior undergraduate level):

LAGA

@book{book:1146567,

title = {Linear and Geometric Algebra},

author = {Alan Macdonald},

publisher = {},

isbn = {1453854932},

year = {2010},

series = {},

edition = {},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=159aca6df4062bdd8c85444df5368476}

}

VAGC

@book{book:1146568,

title = {Vector and Geometric Calculus},

author = {Alan Macdonald},

publisher = {},

isbn = {1480132454},

year = {2012},

series = {},

edition = {},

volume = {},

url = {http://gen.lib.rus.ec/book/index.php?md5=883db09af89649180ce4abea50500b8e}

}

Others on Universal Geometric Algebra and Geometric Calculus if needed:

@book{book:227217,
title = {Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics)},
author = {D. Hestenes, Garret Sobczyk},
publisher = {Springer},
isbn = {9789027725615,9027725616},
year = {1987},
series = {Fundamental Theories of Physics},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=5B4109AECE15D075A4B1DE4B06E389F5}}

@book{book:790059,
title = {A New Approach to Differential Geometry using Clifford’s Geometric Algebra },
author = {John Snygg (auth.)},
publisher = {Birkhäuser Basel},
isbn = {0817682821,9780817682828,9780817682835},
year = {2012},
series = {},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=5DD71F6A00D724E868817B7E616C8F38}
}

@book{book:1351370,
title = {Space-Time Algebra},
author = {David Hestenes; Anthony Lasenby (foreword)},
publisher = {Birkhäuser},
isbn = {3319184121,9783319184128, 9783319184135},
year = {2015},
series = {},
edition = {2},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=b50b8b7703a526812e753cf5352bb7dc}
}

@book{book:246178,
title = {New Foundations for Classical Mechanics (geometric algebra)},
author = {David Hestenes},
publisher = {Kluwer},
isbn = {0-306-47122-1},
year = {2002},
series = {},
edition = {2},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=780479BDEA9E4AF53238809A975E0F61}
}

@book{book:1579005,
title = {Multivectors and Clifford Algebra in Electrodynamics},
author = {Bernard Jancewicz},
publisher = {World Scientific},
isbn = {9971502909,9789971502904},
year = {1989},
series = {},
edition = {},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=ed59213f8779d2d1d21e8ec837073952}
}

@book{book:7403,
title = {Clifford algebras and spinors},
author = {Pertti Lounesto},
publisher = {Cambridge University Press},
isbn = {0521005515,9780521005517},
year = {2001},
series = {London Mathematical Society lecture note series 286},
edition = {2nd ed},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=27B205CDC325106519F7E9D573720A1E}}

@book{book:1392759,
title = {Electrodynamics: A Modern Geometric Approach},
author = {William E. Baylis},
publisher = {Birkhäuser},
isbn = {0-8176-4025-8,3 7643-4025-8},
year = {2002},
series = {},
edition = {},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=90140316ecc4d7b244b3ec20e7877707}
}

For more advanced wider universality:

Hongbo Li-Invariant Algebras And Geometric Reasoning-World Scientific Publishing Company (2008).pdf

@book{book:259944,
title = {Invariant Algebras And Geometric Reasoning},
author = {Hongbo Li},
publisher = {World Scientific Publishing Company},
isbn = {9812708081,9789812708083,9789812770110},
year = {2008},
series = {},
edition = {},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=BEEA3E89DFD6771C7A0C7F9AF61C89D3}
}

Earlier including Tutorial and “Power of positive thinking”:

Wendy Chan, Gian-Carlo Rota, Joel A. Stein (auth.), Neil L. White (eds.)-Invariant Methods in Discrete and Computational Geometry_ Proceedings of the Curaçao Conference, 13–17 June, 1994-Springer Neth.pdf

@book{book:964579,
title = {Invariant Methods in Discrete and Computational Geometry: Proceedings of the Curaçao Conference, 13–17 June, 1994},
author = {Wendy Chan, Gian-Carlo Rota, Joel A. Stein (auth.), Neil L. White (eds.)},
publisher = {Springer Netherlands},
isbn = {978-90-481-4572-0,978-94-015-8402-9},
year = {1995},
series = {},
edition = {1},
volume = {},
url = {http://gen.lib.rus.ec/book/index.php?md5=f72a917d27daa0dfa74fe39985c0ef01}
}

Invariant, or coordinate-free methods provide a natural framework for many geometric questions. Invariant Methods in Discrete andComputational Geometry provides a basic introduction to several aspects of invariant theory, including the supersymmetric algebra, the Grassmann-Cayler algebra, and Chow forms. It also presents a number of current research papers on invariant theory and its applications to problems in geometry, such as automated theorem proving and computer vision.
Audience: Researchers studying mathematics, computers and robotics.

Table of contents :
Front Matter….Pages i-xiii
The Power of Positive Thinking….Pages 1-36
Introduction to Chow Forms….Pages 37-58
Capelli’s Method of Variabili Ausiliarie , Superalgebras and Geometric Calculus….Pages 59-75
Letterplace Algebra and Symmetric Functions….Pages 77-91
A Tutorial on Grassmann-Cayley Algebra….Pages 93-106
Computational Symbolic Geometry….Pages 107-139
Invariant Theory and the Projective Plane….Pages 141-166
Automatic Proving of Geometric Theorems….Pages 167-196
The Resolving Bracket….Pages 197-222
Computation of the Invariants of a Point Set in P 3 from Its Projections in P 2 ….Pages 223-244
Geometric Algebra and Möbius Sphere Geometry as a Basis for Euclidean Invariant Theory….Pages 245-256
Invariants on G/U x G/U x G/U, G = SL(4,C)….Pages 257-277
On a Certain Complex Related to the Notion of Hyperdeterminant….Pages 279-288
On Cayley’s Projective Configurations an Algorithmic Study….Pages 289-299
On the Construction of Equifacetted 3-Spheres….Pages 301-312
Depths and Betti Numbers of Homology Manifolds….Pages 313-321
Back Matter….Pages 323-328

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