Numerical Calculations in Clifford Algebra - Seagar, Andrew

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Résumé :

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 Inverse 106

10.1.8 Reflection 107

10.1.9 Projection 107

10.1.10 Rotation 107

10.2 Practice 107

10.2.1 Example Code 107

10.2.2 Example Output 110

10.3 Field Arithmetic 112

10.3.1 Extensions Based on Quaternions 112

10.3.2 Inverses 113

10.3.3 Example Code 115

10.3.4 Example Output 117

References 118

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