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geometric_algebra [2020/11/19 01:10] – [Articles] pbkgeometric_algebra [2023/12/30 00:23] (current) – [Videos] pbk
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   * [[http://www.cgs-network.org/cgi20|ENGAGE 2020]] (2020) - Empowering Novel Geometric Algebra for Graphics & Engineering Workshop, CGI 2020, Geneva (Switzerland).   * [[http://www.cgs-network.org/cgi20|ENGAGE 2020]] (2020) - Empowering Novel Geometric Algebra for Graphics & Engineering Workshop, CGI 2020, Geneva (Switzerland).
 +
 +  * [[https://bivector.net/game2023.html|GAME2023 - Geometric Algebra Mini Event]] (2023) - DAE - Kortrijk, Belgium.
 ==== Book companion ==== ==== Book companion ====
  
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   * [[http://isites.harvard.edu/fs/docs/icb.topic1048774.files/clif_mein.pdf|Clifford algebras and Lie groups]] {{ga:clif_mein.pdf|(local)}}- //Eckhard Meinrenken//, University of Toronto.   * [[http://isites.harvard.edu/fs/docs/icb.topic1048774.files/clif_mein.pdf|Clifford algebras and Lie groups]] {{ga:clif_mein.pdf|(local)}}- //Eckhard Meinrenken//, University of Toronto.
   * [[https://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009|Applied Geometric Algebra]] (2009) - //László Tisza//, MIT OpenCourseWare.   * [[https://ocw.mit.edu/resources/res-8-001-applied-geometric-algebra-spring-2009|Applied Geometric Algebra]] (2009) - //László Tisza//, MIT OpenCourseWare.
 +  * [[https://www.cefns.nau.edu/~schulz/grassmann.pdf|Theory and application of Grassmann Algebra]] (2011) - //William C. Schulz//, Northern Arizona University.
   * [[https://lyseo.edu.ouka.fi/cms7/sites/default/files/geometric_algebra-tl.pdf|What is geometric algebra?]] (2016) - //Teuvo Laurinolli//, Oulun Lyseon.   * [[https://lyseo.edu.ouka.fi/cms7/sites/default/files/geometric_algebra-tl.pdf|What is geometric algebra?]] (2016) - //Teuvo Laurinolli//, Oulun Lyseon.
   * [[https://www.zatlovac.eu/lecturenotes/GAIntroLagape.pdf|Geometric Algebra and Calculus: Unified Language for Mathematics and Physics]] (2018) - //Vaclav Zatloukal//, Czech Technical University in Prague.   * [[https://www.zatlovac.eu/lecturenotes/GAIntroLagape.pdf|Geometric Algebra and Calculus: Unified Language for Mathematics and Physics]] (2018) - //Vaclav Zatloukal//, Czech Technical University in Prague.
   * [[https://dspace.library.uu.nl/bitstream/handle/1874/383367/IntroductionToGeometricAlgebraV2.pdf|Introduction to Geometric Algebra, a powerful tool for mathematics and physics]] (2019) - //Denis Lamaker//, Universiteit Utrecht.   * [[https://dspace.library.uu.nl/bitstream/handle/1874/383367/IntroductionToGeometricAlgebraV2.pdf|Introduction to Geometric Algebra, a powerful tool for mathematics and physics]] (2019) - //Denis Lamaker//, Universiteit Utrecht.
 +  * [[https://mattferraro.dev/posts/geometric-algebra|What is the Inverse of a Vector?]] (2021) - //Matt Ferraro//.
  
 ===== Videos ===== ===== Videos =====
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   * [[https://www.youtube.com/watch?v=tX4H_ctggYo|SIGGRAPH 2019: Geometric Algebra for Computer Graphics]] - //Charles Gunn// and //Steven De Keninck//.   * [[https://www.youtube.com/watch?v=tX4H_ctggYo|SIGGRAPH 2019: Geometric Algebra for Computer Graphics]] - //Charles Gunn// and //Steven De Keninck//.
   * [[https://www.youtube.com/playlist?list=PLsSPBzvBkYjzcQ4eCVAntETNNVD2d5S79|GAME2020 - Geometric Algebra Mini Event]] - //DAE Kortrijk, Belgium//.   * [[https://www.youtube.com/playlist?list=PLsSPBzvBkYjzcQ4eCVAntETNNVD2d5S79|GAME2020 - Geometric Algebra Mini Event]] - //DAE Kortrijk, Belgium//.
-  * [[https://www.youtube.com/watch?v=60z_hpEAtD8|A Swift Introduction to Geometric Algebra]] - //sudgylacmoe//.+  * [[https://www.youtube.com/watch?v=60z_hpEAtD8|A Swift Introduction to Geometric Algebra]] and [[https://www.youtube.com/playlist?list=PLVuwZXwFua-0Ks3rRS4tIkswgUmDLqqRy|From Zero to Geo]] - //sudgylacmoe//.
   * [[https://www.youtube.com/watch?v=cKfC2ZBJulg|Projective Geometric Algebra for Paraxial Geometric Optics]] - // Katelyn Spadavecchia//.   * [[https://www.youtube.com/watch?v=cKfC2ZBJulg|Projective Geometric Algebra for Paraxial Geometric Optics]] - // Katelyn Spadavecchia//.
 +  * [[https://www.youtube.com/watch?v=11sH9X0OO9Y&list=PLnpuwbuviU2j7OSnZdstP5_g1ejA32bYA|Geometric Algebra Lectures ]] - //Miroslav Josipović//.
 +  * [[https://www.youtube.com/watch?v=HGcBu4TQgRE|Quaternions and Clifford Algebra]] - //Q. J. Ge and Anurag Purwar//, Stony Brook University.
 +  * [[https://www.youtube.com/watch?v=LestlcDk6Iw|Foundations of Geometric Algebra Computing]] - Lecture at ICU Tokyo, //Dietmar Hildenbrand//.
 +  * [[https://www.youtube.com/watch?v=e5D7Bma9Vhw&list=PLxo3PbygE0PLdFFy_2b02JAaUsleFW8py|Geometric Algebra]] - First Course in STEMCstudio, //David Geo Holmes//.
 +  * [[https://www.youtube.com/playlist?list=PLsSPBzvBkYjyWv5wLVV7QfeS_d8pwCPv_|AGACSE2021]] - Selected talks.
 +  * [[https://www.youtube.com/playlist?list=PLsSPBzvBkYjxrsTOr0KLDilkZaw7UE2Vc|Plane-based Geometric Algebra Tutorial]] - Presentation at SIBGRAPI 2021, //Steven De Keninck and Leo Dorst//.
 +  * [[https://www.youtube.com/watch?v=hTa_gErsrtM|Geometric Algebra]] - Talk at SIGGRAPH 2022, //Alyn Rockwood and Dietmar Hildenbrand//.
 +  * [[https://www.youtube.com/watch?v=8n6GsKWznfY|Plane Based Geometric Algebra]] - Advanced Computational Applications of GA, //Leo Dorst and Steven De Keninck//.
 +  * [[https://www.youtube.com/watch?v=PGZNYGwsXTw|Why Geometric Algebra Should be in the Standard Linear Algebra Curriculum]] and [[https://www.youtube.com/watch?v=ISKJPmuZkbY|Fun Applications of Geometric Algebra]] - Presentations at [[https://pgadey.ca/seminar/|Parker Glynn-Adey seminars]], //Logan Lim//.
 +  * [[https://www.youtube.com/watch?v=VXziLgMIWf8|Geometric Clifford Algebra Networks and Clifford Neural Layers for PDE Modeling]] - Valence Labs, //Johannes Brandstetter//.
 +  * [[https://www.youtube.com/watch?v=nktgFWLy32U|Spinors for Beginners 11: What is a Clifford Algebra?]] - //eigenchris//.
 +  * [[https://www.youtube.com/watch?v=htYh-Tq7ZBI|Why can't you multiply vectors?]] - Talk at Dutch Game Day 2023, //Freya Holmér//.
 +  * [[https://www.youtube.com/watch?v=1AmeD0Vc8ow|GAME2023 Geometric Algebra Mini Event]] - Livestream.
 +  * [[https://www.youtube.com/watch?v=zgi-13F2Kec|Geometric Algebra: The Prequel]] - at GAME2023, //Steven De Keninck//.
 +  * [[https://www.youtube.com/watch?v=nPIRL-c88_E|Geometric Algebra Transformers: Revolutionizing Geometric Data]] - at Intelligent Systems Conference 2023, //Taco Cohen//.
 +
 ===== Computing frameworks ===== ===== Computing frameworks =====
   * [[http://www.geometricalgebra.net/downloads.html|GAViewer & GA Sandbox]] - //Leo Dorst, Daniel Fontijne, Stephen Mann//.   * [[http://www.geometricalgebra.net/downloads.html|GAViewer & GA Sandbox]] - //Leo Dorst, Daniel Fontijne, Stephen Mann//.
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   * [[https://arxiv.org/pdf/2007.04464.pdf|Deform, Cut and Tear a skinned model using Conformal Geometric Algebra]] (2020) - //Manos Kamarianakis, George Papagiannakis//   * [[https://arxiv.org/pdf/2007.04464.pdf|Deform, Cut and Tear a skinned model using Conformal Geometric Algebra]] (2020) - //Manos Kamarianakis, George Papagiannakis//
 We present a novel, integrated rigged character simulation framework in Conformal Geometric Algebra (CGA) that supports, for the first time, real-time cuts and tears, before and/or after the animation, while maintaining deformation topology. The purpose of using CGA is to lift several restrictions posed by current state-of-the-art character animation & deformation methods. Previous implementations originally required weighted matrices to perform deformations, whereas, in the current state-of-the-art, dual-quaternions handle both rotations and translations, but cannot handle dilations. CGA is a suitable extension of dual-quaternion algebra that amends these two major previous shortcomings: the need to constantly transmute between matrices and dual-quaternions as well as the inability to properly dilate a model during animation. Our CGA algorithm also provides easy interpolation and application of all deformations in each intermediate steps, all within the same geometric framework. Furthermore we also present two novel algorithms that enable cutting and tearing of the input rigged, animated model, while the output model can be further re-deformed. We present a novel, integrated rigged character simulation framework in Conformal Geometric Algebra (CGA) that supports, for the first time, real-time cuts and tears, before and/or after the animation, while maintaining deformation topology. The purpose of using CGA is to lift several restrictions posed by current state-of-the-art character animation & deformation methods. Previous implementations originally required weighted matrices to perform deformations, whereas, in the current state-of-the-art, dual-quaternions handle both rotations and translations, but cannot handle dilations. CGA is a suitable extension of dual-quaternion algebra that amends these two major previous shortcomings: the need to constantly transmute between matrices and dual-quaternions as well as the inability to properly dilate a model during animation. Our CGA algorithm also provides easy interpolation and application of all deformations in each intermediate steps, all within the same geometric framework. Furthermore we also present two novel algorithms that enable cutting and tearing of the input rigged, animated model, while the output model can be further re-deformed.
 +
 +  * [[https://www.researchgate.net/profile/Carlos-Muro/publication/339249543_Newton-Euler_Modeling_and_Control_of_a_Multi-copter_Using_Motor_Algebra_mathbfG_301G301/links/603d925f299bf1e0784d02bd/Newton-Euler-Modeling-and-Control-of-a-Multi-copter-Using-Motor-Algebra-mathbfG-3-0-1G3-0-1.pdf|Newton-Euler Modelling and Control of a Multicopter using Motor Algebra G^+_3,0,1]] (2020) - //Carlos A. Arellano-Muro, Guillermo Osuna-Gonzalez, et al//
 +In this work the dynamic model and the nonlinear control for a multi-copter have been developed using the geometric algebra framework specifically using the motor algebra G^+_3,0,1. The kinematics for the aircraft model and the dynamics based on Newton-Euler formalism are presented. Block-control technique is applied to the multi-copter model which involves super twisting control and an estimator of the internal dynamics for maneuvers away from the origin. The stability of the presented control scheme is proved. The experimental analysis shows that our non-linear controller law is able to reject external disturbances and to deal with parametric variations.
 +
 +  * [[http://i-us.ru/index.php/ius/article/download/13579/14098|Human action recognition method based on conformal geometric algebra and recurrent neural network]] (2020) - //Nguyen Nang Hung Van, Pham Minh Tuan et al//
 +The use of Conformal Geometric Algebra in order to extract features and simultaneously reduce the dimensionality of a dataset for human activity recognition using Recurrent Neural Network.
 +
 +  * [[https://arxiv.org/pdf/2107.00343.pdf|Explicit Baker-Campbell-Hausdorff-Dynkin formula for Spacetime via Geometric Algebra]] (2021) - //Joseph Wilson, Matt Visser//
 +We present a compact Baker-Campbell-Hausdorff-Dynkin formula for the composition of Lorentz transformations e^σi in the spin representation (a.k.a. Lorentz rotors) in terms of their generators σi. This formula is general to geometric algebras (a.k.a. real Clifford algebras) of dimension ≤4, naturally generalising Rodrigues' formula for rotations in R3. In particular, it applies to Lorentz rotors within the framework of Hestenes' spacetime algebra, and provides an efficient method for composing Lorentz generators.
 +
 +  * [[https://arxiv.org/pdf/2107.03771.pdf|Graded Symmetry Groups: Plane and Simple]] (2021) - //Martin Roelfs, Steven De Keninck//
 +The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the classic matrix Lie algebra approach, while retaining information about the number of reflections in a given transformation. This imposes a graded structure on Lie groups, which is not evident in their matrix representation. By embracing this graded structure, the invariant decomposition theorem was proven: any composition of k linearly independent reflections can be decomposed into ⌈k/2⌉ commuting factors, each of which is the product of at most two reflections. This generalizes a conjecture by M. Riesz, and has e.g. the Mozzi-Chasles' theorem as its 3D Euclidean special case.
 +
 +  * [[https://www.pacm.princeton.edu/sites/default/files/pacm_arjunmani_0.pdf|Representing Words in a Geometric Algebra]] (2021) - //Arjun Mani//
 +In this paper we introduce and motivate geometric algebra as a better representation for word embeddings. Next we describe how to implement the geometric product and interestingly show that neural networks can learn this product. We then introduce a model that represents words as objects in this algebra and benchmark it on large corpuses; our results show some promise on traditional word embedding tasks. Thus, we lay the groundwork for further investigation of geometric algebra in word embeddings.
 +
 +  * [[https://ietresearch.onlinelibrary.wiley.com/doi/pdfdirect/10.1049/cmu2.12188|An approach to adaptive filtering with variable step size based on geometric algebra]] (2021) - //Haiquan Wang, Yinmei He et al//
 +Recently, adaptive filtering algorithms have attracted much more attention in the field of signal processing. By studying the shortcoming of the traditional real-valued fixed step size adaptive filtering algorithm, this paper proposed the novel approach to adaptive filtering with variable step size based on Sigmoid function and geometric algebra (GA).
 +
 +  * [[https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=9488174|A Survey on Quaternion Algebra and Geometric Algebra Applications in Engineering and Computer Science 1995–2020]] (2021) - //Eduardo Bayro-Corrochano//
 +Geometric Algebra (GA) has proven to be an advanced language for mathematics, physics, computer science, and engineering. This review presents a comprehensive study of works on Quaternion Algebra and GA applications in computer science and engineering from 1995 to 2020.
 +
 +  * [[https://dspace.library.uu.nl/bitstream/handle/1874/403340/thesis.pdf|Clifford algebras and their application in the Dirac equation]] (2021) - //Paul van Hoegaerden//
 +The aim of this thesis will be to study the Clifford algebras that appear in the derivation of the Dirac equation and investigate alternative formulations of the Dirac equation using (complex) quaternions. To this end, we will first look at the symmetries of the Dirac equation and some of the additional insights that follow from the Dirac equation. We will also give a derivation of the Dirac equation starting from the Schrödinger equation, in which we will come across the gamma matrices.
 +
 +  * [[https://onlinelibrary.wiley.com/doi/pdfdirect/10.1002/cta.3132|Geometric Algebra for teaching AC Circuit Theory]] (2021) - //Francisco G. Montoya, Raúl Baños et al//
 +This paper presents and discusses the usage of Geometric Algebra (GA) for the analysis of electrical alternating current (AC) circuits. The potential benefits of this novel approach are highlighted in the study of linear and nonlinear circuits with sinusoidal and non-sinusoidal sources in the frequency domain, which are important issues in electrical engineering undergraduate courses.
 ===== Books ===== ===== Books =====
  
 ==== Historical ==== ==== Historical ====
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 ^                                                                                                                                           ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ^ ^                                                                                                                                           ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ^
-| [[https://archive.org/details/dieausdehnungsl04grasgoog|{{:ga:die_ausdehnungslehre_von_1844-grassmann.jpg?400}}]]                         | **Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik [Die Ausdehnungslehre von 1844] (1878)**\\ //Hermann Grassmann//\\ The Prussian schoolmaster Hermann Grassmann taught a range of subjects including mathematics, science and Latin and wrote several secondary-school textbooks. Although he was never appointed to a university post, he devoted much energy to mathematical research and developed revolutionary new insights. Die lineale Ausdehnungslehre, published in 1844, is an astonishing work which was not understood by the mathematicians of its time but which anticipated developments that took a century to come to fruition - vector spaces, dimension, exterior products and many other ideas. Admired rather than read by the next generation, it was only fully appreciated by mathematicians such as Peano and Whitehead.                                                                                                                                                |+| [[https://archive.org/details/dieausdehnungsl04grasgoog|{{:ga:die_ausdehnungslehre_von_1844-grassmann.jpg?100}}]]                         | **Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik [Die Ausdehnungslehre von 1844] (1878)**\\ //Hermann Grassmann//\\ The Prussian schoolmaster Hermann Grassmann taught a range of subjects including mathematics, science and Latin and wrote several secondary-school textbooks. Although he was never appointed to a university post, he devoted much energy to mathematical research and developed revolutionary new insights. Die lineale Ausdehnungslehre, published in 1844, is an astonishing work which was not understood by the mathematicians of its time but which anticipated developments that took a century to come to fruition - vector spaces, dimension, exterior products and many other ideas. Admired rather than read by the next generation, it was only fully appreciated by mathematicians such as Peano and Whitehead.                                                                                                                                                |
 | [[https://archive.org/details/dieausdehnugsle00grasgoog|{{:ga:die_ausdehnungslehre-grassmann.jpg?100}}]]                                  | **Die Ausdehnungslehre (1864)**\\ //Hermann Grassmann//\\ In 1844, the Prussian schoolmaster Hermann Grassmann published Die Lineale Ausdehnungslehre. This revolutionary work anticipated the modern theory of vector spaces and exterior algebras. It was little understood at the time and the few sympathetic mathematicians, rather than trying harder to comprehend it, urged Grassmann to write an extended version of his theories. The present work is that version, first published in 1862. However, this also proved too far ahead of its time and Grassmann turned to historical linguistics, in which field his contributions are still remembered. His mathematical work eventually found champions such as Hankel, Peano, Whitehead and Élie Cartan, and it is now recognised for the brilliant achievement that it was in the history of mathematics.                                                                                                                                        | | [[https://archive.org/details/dieausdehnugsle00grasgoog|{{:ga:die_ausdehnungslehre-grassmann.jpg?100}}]]                                  | **Die Ausdehnungslehre (1864)**\\ //Hermann Grassmann//\\ In 1844, the Prussian schoolmaster Hermann Grassmann published Die Lineale Ausdehnungslehre. This revolutionary work anticipated the modern theory of vector spaces and exterior algebras. It was little understood at the time and the few sympathetic mathematicians, rather than trying harder to comprehend it, urged Grassmann to write an extended version of his theories. The present work is that version, first published in 1862. However, this also proved too far ahead of its time and Grassmann turned to historical linguistics, in which field his contributions are still remembered. His mathematical work eventually found champions such as Hankel, Peano, Whitehead and Élie Cartan, and it is now recognised for the brilliant achievement that it was in the history of mathematics.                                                                                                                                        |
 | [[https://archive.org/details/bub_gb_bU9rkSdWlFAC|{{:ga:theorie_der_complexen_zahlensysteme-hankel.jpg?100}}]]                            | **Theorie Der Complexen Zahlensysteme (1867)**\\ //Hermann Hankel//\\ Insbesondere der Gemeinen Imaginären Zahlen und der Hamilton'schen Quaternionen Nebst Ihrer Geometrischen Darstellung.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | | [[https://archive.org/details/bub_gb_bU9rkSdWlFAC|{{:ga:theorie_der_complexen_zahlensysteme-hankel.jpg?100}}]]                            | **Theorie Der Complexen Zahlensysteme (1867)**\\ //Hermann Hankel//\\ Insbesondere der Gemeinen Imaginären Zahlen und der Hamilton'schen Quaternionen Nebst Ihrer Geometrischen Darstellung.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
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 ==== Modern ==== ==== Modern ====
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 ^                                                                                                                                                                                                         ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ^ ^                                                                                                                                                                                                         ^ Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         ^
-| [[https://www.amazon.com/Space-Time-Algebra-David-Hestenes/dp/3319184121|{{:ga:space-time_algebra-hestenes.jpg?650}}]] | **Space-Time Algebra, 2nd Ed (2015)**\\ //David Hestenes//\\ This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future. At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |+| [[https://www.amazon.com/Space-Time-Algebra-David-Hestenes/dp/3319184121|{{:ga:space-time_algebra-hestenes.jpg?100}}]] | **Space-Time Algebra, 2nd Ed (2015)**\\ //David Hestenes//\\ This small book started a profound revolution in the development of mathematical physics, one which has reached many working physicists already, and which stands poised to bring about far-reaching change in the future. At its heart is the use of Clifford algebra to unify otherwise disparate mathematical languages, particularly those of spinors, quaternions, tensors and differential forms. It provides a unified approach covering all these areas and thus leads to a very efficient ‘toolkit’ for use in physical problems including quantum mechanics, classical mechanics, electromagnetism and relativity (both special and general) – only one mathematical system needs to be learned and understood, and one can use it at levels which extend right through to current research topics in each of these areas.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
 | [[https://www.amazon.com/Foundations-Classical-Mechanics-Fundamental-Theories/dp/0792355148|{{:ga:new_foundations_for_classical_mechanics-hestenes.jpg?100}}]] | **New Foundations for Classical Mechanics, 2nd Ed (1999)**\\ //David Hestenes//\\ This is a textbook on classical mechanics at the intermediate level, but its main purpose is to serve as an introduction to a new mathematical language for physics called geometric algebra. Mechanics is most commonly formulated today in terms of the vector algebra developed by the American physicist J. Willard Gibbs, but for some applications of mechanics the algebra of complex numbers is more efficient than vector algebra, while in other applications matrix algebra works better. Geometric algebra integrates all these algebraic systems into a coherent mathematical language which not only retains the advantages of each special algebra but possesses powerful new capabilities. This book covers the fairly standard material for a course on the mechanics of particles and rigid bodies. However, it will be seen that geometric algebra brings new insights into the treatment of nearly every topic and produces simplifications that move the subject quickly to advanced levels.                                                                                                                                                                                                                                                                                 | | [[https://www.amazon.com/Foundations-Classical-Mechanics-Fundamental-Theories/dp/0792355148|{{:ga:new_foundations_for_classical_mechanics-hestenes.jpg?100}}]] | **New Foundations for Classical Mechanics, 2nd Ed (1999)**\\ //David Hestenes//\\ This is a textbook on classical mechanics at the intermediate level, but its main purpose is to serve as an introduction to a new mathematical language for physics called geometric algebra. Mechanics is most commonly formulated today in terms of the vector algebra developed by the American physicist J. Willard Gibbs, but for some applications of mechanics the algebra of complex numbers is more efficient than vector algebra, while in other applications matrix algebra works better. Geometric algebra integrates all these algebraic systems into a coherent mathematical language which not only retains the advantages of each special algebra but possesses powerful new capabilities. This book covers the fairly standard material for a course on the mechanics of particles and rigid bodies. However, it will be seen that geometric algebra brings new insights into the treatment of nearly every topic and produces simplifications that move the subject quickly to advanced levels.                                                                                                                                                                                                                                                                                 |
 | [[https://www.amazon.com/Clifford-Algebra-Geometric-Calculus-Mathematics/dp/9027725616|{{:ga:clifford_algebra_to_geometric_calculus-hestenes_sobczyk.jpg?100}}]] | **Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (1984)**\\ //David Hestenes, Garret Sobczyk//\\ Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quatemions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.                                                            | | [[https://www.amazon.com/Clifford-Algebra-Geometric-Calculus-Mathematics/dp/9027725616|{{:ga:clifford_algebra_to_geometric_calculus-hestenes_sobczyk.jpg?100}}]] | **Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (1984)**\\ //David Hestenes, Garret Sobczyk//\\ Matrix algebra has been called "the arithmetic of higher mathematics" [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called 'Clifford Algebra', though we prefer the name 'Geometric Algebm' suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quatemions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.                                                            |
geometric_algebra.1605748219.txt.gz · Last modified: 2020/11/19 01:10 by pbk

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