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Geometric Multiplication of Vectors: An Introduction to Geometric Algebra in Physics 2019 ed. [Minkštas viršelis]

  • Formatas: Paperback / softback, 241 pages, aukštis x plotis: 235x155 mm, weight: 454 g, 1 Illustrations, color; 88 Illustrations, black and white; XXV, 241 p. 89 illus., 1 illus. in color., 1 Paperback / softback
  • Serija: Compact Textbooks in Mathematics
  • Išleidimo metai: 03-Dec-2019
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030017559
  • ISBN-13: 9783030017552
Kitos knygos pagal šią temą:
Geometric Multiplication of Vectors: An Introduction to Geometric Algebra in Physics 2019 ed.Didesnis paveikslėlis
  • Formatas: Paperback / softback, 241 pages, aukštis x plotis: 235x155 mm, weight: 454 g, 1 Illustrations, color; 88 Illustrations, black and white; XXV, 241 p. 89 illus., 1 illus. in color., 1 Paperback / softback
  • Serija: Compact Textbooks in Mathematics
  • Išleidimo metai: 03-Dec-2019
  • Leidėjas: Springer Nature Switzerland AG
  • ISBN-10: 3030017559
  • ISBN-13: 9783030017552
Kitos knygos pagal šią temą:
Pastovi nuoroda: https://www.kriso.lt/db/9783030017552.html

This book enables the reader to discover elementary concepts of geometric algebra and its applications with lucid and direct explanations. Why would one want to explore geometric algebra? What if there existed a universal mathematical language that allowed one: to make rotations in any dimension with simple formulas, to see spinors or the Pauli matrices and their products, to solve problems of the special theory of relativity in three-dimensional Euclidean space, to formulate quantum mechanics without the imaginary unit, to easily solve difficult problems of electromagnetism, to treat the Kepler problem with the formulas for a harmonic oscillator, to eliminate unintuitive matrices and tensors, to unite many branches of mathematical physics? What if it were possible to use that same framework to generalize the complex numbers or fractals to any dimension, to play with geometry on a computer, as well as to make calculations in robotics, ray-tracing and brain science? In addition, what if such a language provided a clear, geometric interpretation of mathematical objects, even for the imaginary unit in quantum mechanics? Such a mathematical language exists and it is called geometric algebra. High school students have the potential to explore it, and undergraduate students can master it. The universality, the clear geometric interpretation, the power of generalizations to any dimension, the new insights into known theories, and the possibility of computer implementations make geometric algebra a thrilling field to unearth.


1 Basic Concepts
1(74)
1.1 Geometric Product
1(14)
1.1.1 How to Multiply Vectors
2(2)
1.1.2 Grassmann's Great Idea
4(3)
1.1.3 The Symmetric and Antisymmetric Parts of the Product
7(1)
1.1.4 Orthonormal Basis Vectors
8(2)
1.1.5 The Inner and the Outer Products
10(1)
1.1.6 Graphical Representation of Bivectors
11(2)
1.1.7 Equations of Geometric Objects
13(1)
1.1.8 The Lorentz Force and Space Inversion
14(1)
1.2 Complex Numbers
15(1)
1.3 The Cross Product and Duality
16(1)
1.3.1 Duality Operation
17(1)
1.4 Algebra
17(3)
1.4.1 Grades and Parity of a Multivector
19(1)
1.4.2 Projection and Rejection
20(1)
1.5 Mixed Products
20(1)
1.5.1 A Useful Formula
21(1)
1.6 Important Concepts
21(6)
1.6.1 Versor
21(1)
1.6.2 Blade
21(1)
1.6.3 Involutions
22(2)
1.6.4 Inverses
24(1)
1.6.5 Nilpotents
24(1)
1.6.6 Idempotents
25(1)
1.6.7 Zero Divisors
25(1)
1.6.8 Addition of Different Grades
25(1)
1.6.9 Lists of Coefficients
26(1)
1.7 Examples of Solving Equations
27(1)
1.7.1 Quadratic Equations
27(1)
1.8 Geometric Product of Vectors in Trigonometric Form
28(1)
1.8.1 Merging Multiplication Tables
29(1)
1.9 Reflections, Rotations, Spinors, Quaternions
29(18)
1.9.1 Bivectors as Rotors
29(3)
1.9.2 Invariants of Rotations
32(1)
1.9.3 The Formalism of Reflections and Rotations
33(2)
1.9.4 A Special Rotor Construction
35(1)
1.9.5 Rotors as Geometric Products of Two Unit Vectors
36(2)
1.9.6 Small Rotations
38(1)
1.9.7 Rotors and Eigenblades
39(1)
1.9.8 Rotors Are Strange
39(2)
1.9.9 How to Rotate a Basis
41(1)
1.9.10 Rotors and Quaternions
41(1)
1.9.11 Rotors Are Universal
42(1)
1.9.12 Rotors as Arcs
42(1)
1.9.13 Rotors Are Unit Spinors
43(1)
1.9.14 Angles as Areas
43(1)
1.9.15 The Versor Product
44(1)
1.9.16 Special Versors
45(2)
1.10 The Scalar Product and the Magnitude of Multivectors
47(2)
1.10.1 The Multivector Norm
48(1)
1.11 Contractions
49(7)
1.11.1 Left Contractions and Permutations
51(1)
1.11.2 Left Contraction and k-Vectors
52(1)
1.11.3 Projectors
53(2)
1.11.4 A Proof of a Special Formula
55(1)
1.12 Commutators and Orthogonal Transformations
56(2)
1.12.1 Commutators with Bivectors
56(2)
1.13 Functions of a Complex Argument
58(2)
1.13.1 The Cauchy--Riemann Equations
58(1)
1.13.2 Taylor Expansion and Analytic Functions
59(1)
1.14 Spinors
60(1)
1.15 A Bit of "Ordinary" Physics
61(3)
1.15.1 The Kepler Problem
62(2)
1.16 Words and Sentences
64(2)
1.16.1 Geometric Content of Expressions
65(1)
1.17 Linear Transformations
66(9)
1.17.1 Determinants
70(2)
1.17.2 Inverse of a Linear Transformation
72(1)
1.17.3 Examples of Linear Transformations
72(1)
1.17.4 Eigenvectors and Eigenblades
73(2)
2 Euclidean 3D Geometric Algebra (C/3)
75(46)
2.1 The Structure of Multivectors in C/3
75(4)
2.1.1 The Square of a Complex Vector
78(1)
2.2 Nilpotents and Dual Numbers
79(3)
2.2.1 Dual Numbers
80(1)
2.2.2 Dual Numbers and Galilean Transformations
81(1)
2.3 Idempotents and Hyperbolic Structure
82(1)
2.4 Spectral Decomposition and Functions of Multivectors
83(9)
2.4.1 Motivation
83(1)
2.4.2 Spectral Decomposition
84(1)
2.4.3 Series Expansion
85(2)
2.4.4 Functions of Complex Vectors
87(1)
2.4.5 Functions of Nilpotents
87(1)
2.4.6 Functions of Idempotents and Unit Complex Vectors
87(1)
2.4.7 Inverse Functions
88(1)
2.4.8 Functions of Lightlike Multivectors
88(1)
2.4.9 Elementary Functions
89(2)
2.4.10 Polynomial Equations
91(1)
2.4.11 A New Concept of Numbers
91(1)
2.5 What Is the Square Root of --1?
92(1)
2.6 Trigonometric Forms of Multivectors
93(3)
2.6.1 Mathematics of the Special Theory of Relativity
94(1)
2.6.2 Everything Is a "Boost"
95(1)
2.7 The Special Theory of Relativity
96(12)
2.7.1 Postulates of the Special Theory of Relativity
97(1)
2.7.2 Inertial Coordinate Systems and Reference Frames
97(1)
2.7.3 Derivation of the Lorentz Transformations from Symmetries
98(1)
2.7.4 Paravectors and Restricted Lorentz Transformations
99(3)
2.7.5 Paravectors and the Minkowski Metric
102(2)
2.7.6 The Restricted Lorentz Transformations
104(1)
2.7.7 Electromagnetic Fields in Geometric Algebra
105(1)
2.7.8 Problems with the Cross Product
106(1)
2.7.9 Complex Vectors Are Powerful
107(1)
2.8 Eigenspinors
108(2)
2.9 Spinorial Equations
110(1)
2.9.1 Kepler's Problem
111(1)
2.10 C/3 and Quantum Mechanics
111(5)
2.10.1 The Wave Function of the Electron. Spinors
112(1)
2.10.2 Spinors in C/3
113(1)
2.10.3 Analogies for the Action of Operators
114(1)
2.10.4 Observables in the Pauli Theory
114(1)
2.10.5 The Expected Value of the Spin
115(1)
2.10.6 Spinors Are Rotors with Dilatation
115(1)
2.10.7 Half Spin is Due to the 3D Geometry
116(1)
2.11 Differentiation and Integration
116(1)
2.12 Geometric Models (The Conformal Model)
117(4)
2.12.1 Points as Null Vectors
118(1)
2.12.2 Geometric Objects
118(3)
3 Applications
121(20)
3.1 Two Boosts
121(3)
3.2 Two Rotors in 3D
124(1)
3.3 Reciprocal Frames
125(4)
3.3.1 An Example of a Reciprocal Frame
127(2)
3.4 Rigid-Body Dynamics
129(3)
3.5 C/2 and Complex Numbers
132(3)
3.5.1 Inversion
134(1)
3.6 Generalization of Real and Complex Products of Complex Numbers
135(1)
3.7 The Complex Geometric Product and Fractals
135(1)
3.8 Multiplication of Blades and Programming
136(5)
3.8.1 An Interesting Mathematica Implementation
137(1)
3.8.2 Bits and Logical Operations
138(1)
3.8.3 Multiplication Tables
138(1)
3.8.4 Implementation of C/3 via Spectral Decomposition
139(2)
4 Geometric Algebra and Matrices
141(20)
4.1 The Pauli Matrices
142(3)
4.1.1 Famous Representation of Vectors
142(1)
4.1.2 Some Properties of the Pauli Matrices
143(2)
4.1.3 The Pauli Matrices and Relativity
145(1)
4.2 Spectral Basis and Matrices
145(6)
4.2.1 C/2
145(4)
4.2.2 C/3
149(2)
4.3 Zero Divisors and Cancellation of Factors
151(1)
4.4 Classical Spinors and Matrices
152(8)
4.4.1 Spinor Formula
153(2)
4.4.2 Exponential of a Matrix
155(1)
4.4.3 SO(3) and SU(2)
156(1)
4.4.4 Spin Rotation Matrices and Vectors
157(1)
4.4.5 Spinors in the Special Theory of Relativity
157(1)
4.4.6 Theorems About Determinants
158(1)
4.4.7 SL(2,C)
158(1)
4.4.8 Generators of the Lorentz Group
159(1)
4.5 Vahlen Matrices and Mobius Transformations
160(1)
5 Appendix
161(28)
5.1 The Exponential Function
161(3)
5.2 Products as Sums
164(1)
5.3 On Idempotents and Spinors
165(5)
5.3.1 Idempotents
165(4)
5.3.2 Bases for Spinorial Matrices
169(1)
5.3.3 Observables in Quantum Mechanics
169(1)
5.3.4 The Mobius Strip and Spinors
170(1)
5.4 Extended Lorentz Transformations. The Speed Limit?
170(7)
5.4.1 General Bilinear Transformations
171(1)
5.4.2 The Real Proper Time
172(1)
5.4.3 The New Limiting Speed(s)
173(1)
5.4.4 The Generalized Velocity and the Generalized Momentum
174(1)
5.4.5 New Conserved Quantities
174(1)
5.4.6 Properties of General Transformations
175(1)
5.4.7 Maxwell's Equations Under General Transformations
175(2)
5.5 Visualization of the Electromagnetic Field in Vacuum
177(2)
5.6 Paravectors and EM Fields
179(2)
5.7 Eigenvalues and Eigenelements
181(3)
5.7.1 Eigensystem of Elements from the Clifford Basis
182(2)
5.8 Duality
184(1)
5.9 Permutations of Orthonormal Unit Vectors in C/3
185(4)
6 Miscellaneous
189
6.1 Solutions to Selected Problems
189(18)
6.2 Problems
207(10)
6.3 Why Geometric Algebra?
217(3)
6.4 Formulas
220
6.4.1 Linear Transformations
222(1)
6.4.2 C/3
222(1)
6.4.3 The Special Theory of Relativity
223(1)
6.4.4 Lorentz Transformations
224(1)
6.4.5 Electromagnetic Field
224(1)
6.4.6 Quantum Mechanics
225(1)
6.4.7 Conformal Model (Hestenes)
225
Correction to: Geometric Multiplication of Vectors 1(226)
How to Proceed Further? 227(2)
Quotes 229(2)
Credits 231(2)
Glossary 233(4)
Literature 237(1)
Some More Interesting Texts 238(1)
References and Links to Specific Subjects 239(1)
References 239(1)
Links to Supplementary Materials 239(1)
3D Geometry 239(1)
Some Web Resources 239(1)
Software 240
Miroslav Josipovi is a Croatian physicist and musician with special interest in revising the language of mathematical physics and in creating new ways of teaching physics.