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What is included with this book?
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:
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.
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.
Preface xix
I Entities and Operations
1 Introduction 3
1.1 Operations 3
1.2 History 4
1.3 Alternative Forms 4
1.4 Naming 5
1.5 Structure 6
1.5.1 Algebraic 6
1.5.2 Numeric 6
1.6 Entities 9
2 Input 11
2.1 Syntax 11
2.2 Constants 12
2.2.1 Specific Types 12
2.2.2 General 13
2.3 Variables 15
2.3.1 Checking and Converting 15
3 Output 21
3.1 Tree Format 21
3.2 Numeric Formats 24
3.2.1 Default Format 24
3.2.2 Defined Format 25
3.3 Extended Formats 26
3.3.1 Rounding 26
v
vi c Seagar/Wiley, September 16, 2022 Contents
3.3.2 Parts of Coefficients 27
3.4 Selected Components 28
3.5 Primitive Formats 29
3.6 Recovered Values 30
4 Unary Operations 33
4.1 Theory 33
4.1.1 Negation 33
4.1.2 Involution 34
4.1.3 Pair Exchange 34
4.1.4 Reversion 35
4.1.5 Clifford Conjugation 36
4.1.6 Supplementation and Pseudoscalar 36
4.2 Practice 37
4.2.1 Example Code 37
4.2.2 Example Output 38
5 Binary Operations 41
5.1 Geometric Origins 41
5.1.1 Outer Multiplication 41
5.1.2 Orthogonal Components 43
5.1.3 Inner Multiplication 44
5.1.4 Names 45
5.2 Multiplication of Units 46
5.2.1 Progressive and Regressive Multiplication 46
5.2.2 Outer, Inner and Central Multiplication 47
5.2.3 Multiplication by Scalars 49
5.3 Central Multiplication 49
5.3.1 Primal Units 50
5.3.2 Evolved and Other Units 51
5.3.3 Numbers 52
5.4 Practice 53
5.4.1 Example Code 53
5.4.2 Example Output 54
5.4.3 Multiplication Tables 54
Contents c Seagar/Wiley, September 16, 2022 vii
6 Vectors and Geometry 59
6.1 Theory 59
6.1.1 Magnitude 59
6.1.2 Inverse 60
6.1.3 Reflection 60
6.1.4 Projection 60
6.1.5 Rotation 61
6.2 Practice 61
6.2.1 Example Code 61
6.2.2 Example Output 63
7 Quaternions 65
7.1 Theory 65
7.1.1 Magnitude 65
7.1.2 Inverse 66
7.1.3 Reflection and Projection 66
7.1.4 Rotation 66
7.1.5 Intersection 67
7.1.6 Factorisation 67
7.2 Practice 68
7.2.1 Example Code 68
7.2.2 Example Output 70
8 Pauli Matrices 73
8.1 Theory 73
8.1.1 Recovery of Components 73
8.1.2 Magnitude 74
8.1.3 Inverse 74
8.1.4 Reflection, Projection and Rotation 74
8.2 Practice 74
8.2.1 Example Code 74
8.2.2 Example Output 76
9 Bicomplex Numbers 79
9.1 Theory 79
9.1.1 Conjugate 79
9.1.2 Magnitude 80
viii c Seagar/Wiley, September 16, 2022 Contents
9.1.3 Inverse 80
9.1.4 Reflection, Projection, and Rotation 80
9.2 Practice 81
9.2.1 Example Code 81
9.2.2 Example Output 82
10 Electromagnetic Fields 85
10.1 Theory 85
10.1.1 Time and Frequency 85
10.1.2 Electromagnetic Entities 85
10.1.3 Dirac Operators 86
10.1.4 Maxwell’s Equations 86
10.1.5 Simplified Notation 87
10.1.6 Magnitude 87
10.1.7 Inverse 87
10.1.8 Reflection 88
10.1.9 Projection 88
10.1.10 Rotation 88
10.2 Practice 88
10.2.1 Example Code 88
10.2.2 Example Output 90
10.3 Field Arithmetic 92
10.3.1 Extensions based on Quaternions 92
10.3.2 Inverses 92
10.3.3 Example Code 94
10.3.4 Example Output 95
11 Arrays of Clifford Numbers 97
11.1 Theory 97
11.2 Practice 98
11.2.1 Example Code 98
11.2.2 Example Output 100
12 Power Series 103
12.1 Theory 103
12.1.1 User Defined 103
12.1.2 Predefined 104
Contents c Seagar/Wiley, September 16, 2022 ix
12.1.3 Convergence 105
12.1.4 Factorisation 105
12.1.5 Squaring 106
12.2 Practice 106
12.2.1 User Defined 106
12.2.2 Predefined 109
13 Matrices of Clifford Numbers 121
13.1 Background 121
13.2 Inversion 122
13.3 Practice 123
13.3.1 Example Code 123
13.3.2 Example Output 126
II Customisation
14 Memory 133
14.1 Memory Usage 133
14.2 Examples 134
14.2.1 Memory Tree Sparsity 134
14.2.2 Memory Expansion 138
14.2.3 Memory Recycling 139
15 Errors 143
15.1 User Errors 143
15.1.1 Syntax Errors and Messages 146
15.2 System Errors 146
15.3 Recovery 146
15.4 Beneficial Usage 150
16 Extension 155
16.1 Accumulation 155
16.2 Multiplication 156
16.3 Transformation 158
16.4 Filtration 159
x c Seagar/Wiley, September 16, 2022 Contents
III Application
17 Verification 165
17.1 Identities 165
17.2 Tests 165
17.2.1 Example Code 165
17.2.2 Example Output 167
18 Lines Not Parallel 173
18.1 Theory 173
18.1.1 Common Plane 173
18.1.2 No Plane in Common 175
18.2 Practice 177
18.2.1 Example Code 177
18.2.2 Example Output 180
19 Perspective Projection 181
19.1 Theory 181
19.2 Practice 182
19.2.1 Example Code 182
19.2.2 Example Output 185
20 Linear Systems 187
20.1 Theory 187
20.2 Practice 188
20.2.1 Example Code 188
20.2.2 Example Output 190
21 Fast Fourier Transform 191
21.1 Theory 191
21.2 Practice 192
21.2.1 Example Code 192
21.2.2 Example Output 196
22 Hertzian Dipole 199
22.1 Theory 199
22.2 Practice 200
22.2.1 Example Code 200
22.2.2 Example Output 203
Contents c Seagar/Wiley, September 16, 2022 xi
23 Finite Difference Time Domain 207
23.1 Theory 207
23.1.1 Analytical Solution 207
23.1.2 Series Solution 207
23.1.3 Analytical Example 208
23.1.4 Numerical Derivatives 209
23.2 Practice 210
23.2.1 Example Code 210
23.2.2 Example Output 215
24 Cauchy Extension 221
24.1 Background 221
24.2 Theory 221
24.2.1 Two Dimensions 221
24.2.2 Three Dimensions 222
24.2.3 Singularity 222
24.2.4 The Taming Function 223
24.2.5 Construction 223
24.3 Practice 225
24.3.1 Example Code 225
24.3.2 Example Output 228
25 Electromagnetic Scattering 233
25.1 Background 233
25.2 Theory 234
25.3 Practice 235
25.3.1 Example Code 235
25.3.2 Example Output 236
IV Programming
26 Interfaces 243
26.1 Configuration and Observation 243
26.1.1 Management 243
26.1.2 Printing 244
26.2 Simple Entities 245
26.2.1 Units 245
xii c Seagar/Wiley, September 16, 2022 Contents
26.2.2 Components 245
26.2.3 Numbers 247
26.3 Higher Entities 249
26.3.1 Vectors 249
26.3.2 Bicomplex Numbers 250
26.3.3 Quaternions 250
26.3.4 Pauli Matrices 251
26.3.5 Electromagnetic Fields 251
26.4 Multiple Entities 251
26.4.1 Arrays 251
26.4.2 Fast Fourier Transforms 252
26.4.3 Series 252
26.4.4 Matrices 253
27 Descriptions 255
27.1 Arguments 255
27.2 Datatypes 255
27.3 Formats 257
27.4 Manual Pages 259
27.4.1 A–E 259
27.4.2 F–J 276
27.4.3 K–O 294
27.4.4 P–T 306
27.4.5 U–Z 360
27.5 Quick Reference 367
Appendices 375
A Key to Example Code and Results 375
Index 377
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