Andrew Seagar

Numerical Calculations in Clifford Algebra (eBook, ePUB)

A Practical Guide for Engineers and Scientists

• Format: ePub

Produktbeschreibung

NUMERICAL CALCULATIONS IN CLIFFORD ALGEBRA An intuitive combination of the theory of Clifford algebra with numerous worked and computed examples and calculations Numerical Calculations in Clifford Algebra: A Practical Guide for Engineers and Scientists is an accessible and practical introduction to Clifford algebra, with comprehensive coverage of the theory and calculations. The book offers many worked and computed examples at a variety of levels of complexity and over a range of different applications making extensive use of diagrams to maintain clarity. The author introduces and documents the Clifford Numerical Suite, developed to overcome the limitations of existing computational packages and to enable the rapid creation and deployment of sophisticated and efficient code. Applications of the suite include Fourier transforms for arrays of any types of Clifford numbers and the solution of linear systems in which the coefficients are Clifford numbers of particular types, including scalars, bicomplex numbers, quaternions, Pauli matrices, and extended electromagnetic fields. Readers will find: * A thorough introduction to Clifford algebra, with a combination of theory and practical implementation in a range of engineering problems * Comprehensive explorations of a variety of worked and computed examples at various levels of complexity * Practical discussions of the conceptual and computational tools for solving common engineering problems * Detailed documentation on the deployment and application of the Clifford Numerical Suite Perfect for engineers, researchers, and academics with an interest in Clifford algebra, Numerical Calculations in Clifford Algebra: A Practical Guide for Engineers and Scientists will particularly benefit professionals in the areas of antenna design, digital image processing, theoretical physics, and geometry.

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Produktdetails

• Verlag: John Wiley & Sons
• Seitenzahl: 528
• Erscheinungstermin: 25. Mai 2023
• Englisch
• ISBN-13: 9781394173259
• Artikelnr.: 68165110

Autorenporträt

Andrew Seagar, PhD, is Director for the Bachelor of Engineering Programs at the Gold Coast Campus of the School of Engineering at Griffith University in Australia. He has experience in a variety of research, commercial development, and academic positions around the world, primarily in the areas of electrical or biomedical engineering.

Inhaltsangabe

List of Figures xv

List of Tables xix

Preface xxi

Part I Entities and Operations 1

1 Introduction 3

1.1 Operations 3

1.2 History 4

1.3 Alternative Forms 5

1.4 Naming 6

1.5 Structure 7

1.5.1 Algebraic 7

1.5.2 Numeric 8

1.6 Entities 11

References 12

2 Input 13

2.1 Syntax 13

2.2 Constants 14

2.2.1 Specific Types 14

2.2.2 General 16

2.3 Variables 19

2.3.1 Checking and Converting 19

Reference 23

3 Output 25

3.1 Tree Format 26

3.2 Numeric Formats 29

3.2.1 Default Format 29

3.2.2 Defined Format 31

3.3 Extended Formats 32

3.3.1 Rounding 32

3.3.2 Parts of Coefficients 33

3.4 Selected Components 35

3.5 Primitive Formats 36

3.6 Recovered Values 38

4 Unary Operations 41

4.1 Theory 41

4.1.1 Negation 41

4.1.2 Involution 41

4.1.3 Pair Exchange 42

4.1.4 Reversion 43

4.1.5 Clifford Conjugation 44

4.1.6 Supplementation and Pseudo-scalar 44

4.2 Practice 45

4.2.1 Example Code 45

4.2.2 Example Output 47

5 Binary Operations 49

5.1 Geometric Origins 49

5.1.1 Outer Multiplication 49

5.1.2 Orthogonal Components 52

5.1.3 Inner Multiplication 53

5.1.4 Names 54

5.2 Multiplication of Units 55

5.2.1 Progressive and Regressive Multiplication 55

5.2.2 Outer, Inner, and Central Multiplication 57

5.2.3 Multiplication By Scalars 58

5.3 Central Multiplication 59

5.3.1 Primal Units 60

5.3.2 Evolved and Other Units 61

5.3.3 Numbers 62

5.4 Practice 63

5.4.1 Example Code 63

5.4.2 Example Output 65

5.4.3 Multiplication Tables 65

References 70

6 Vectors and Geometry 71

6.1 Theory 71

6.1.1 Magnitude 71

6.1.2 Inverse 72

6.1.3 Reflection 72

6.1.4 Projection 73

6.1.5 Rotation 73

6.2 Practice 74

6.2.1 Example Code 74

6.2.2 Example Output 76

7 Quaternions 79

7.1 Theory 79

7.1.1 Magnitude 80

7.1.2 Inverse 80

7.1.3 Reflection and Projection 80

7.1.4 Rotation 81

7.1.5 Intersection 82

7.1.6 Factorisation 82

7.2 Practice 83

7.2.1 Example Code 83

7.2.2 Example Output 86

References 87

8 Pauli Matrices 89

8.1 Theory 89

8.1.1 Recovery of Components 90

8.1.2 Magnitude 90

8.1.3 Inverse 91

8.1.4 Reflection, Projection, and Rotation 91

8.2 Practice 91

8.2.1 Example Code 91

8.2.2 Example Output 94

Reference 95

9 Bicomplex Numbers 97

9.1 Theory 97

9.1.1 Conjugate 98

9.1.2 Magnitude 98

9.1.3 Inverse 98

9.1.4 Reflection, Projection, and Rotation 99

9.2 Practice 99

9.2.1 Example Code 99

9.2.2 Example Output 101

Reference 102

10 Electromagnetic Fields 103

10.1 Theory 103

10.1.1 Time and Frequency 103

10.1.2 Electromagnetic Entities 104

10.1.3 Dirac Operators 105

10.1.4 Maxwell's Equations 105

10.1.5 Simplified Notation 105

10.1.6 Magnitude 106

10.1.7 I