Understanding Geometric Algebra for Electromagnetic Theory
by Arthur, John W.Buy New
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Summary
Author Biography
Table of Contents
| Preface | p. xi |
| Reading Guide | p. xv |
| Introduction | p. 1 |
| A Quick Tour of Geometric Algebra | p. 7 |
| The Basic Rules of a Geometric Algebra | p. 16 |
| 3D Geometric Algebra | p. 17 |
| Developing the Rules | p. 19 |
| General Rules | p. 20 |
| 3D | p. 21 |
| The Geometric Interpretation of Inner and Outer Products | p. 22 |
| Comparison with Traditional 3D Tools | p. 24 |
| New Possibilities | p. 24 |
| Exercises | p. 26 |
| Applying the Abstraction | p. 27 |
| Space and Time | p. 27 |
| Electromagnetics | p. 28 |
| The Electromagnetic Field | p. 28 |
| Electric and Magnetic Dipoles | p. 30 |
| The Vector Derivative | p. 32 |
| The Integral Equations | p. 34 |
| The Role of the Dual | p. 36 |
| Exercises | p. 37 |
| Generalization | p. 39 |
| Homogeneous and Inhomogeneous Multivectors | p. 40 |
| Blades | p. 40 |
| Reversal | p. 42 |
| Maximum Grade | p. 43 |
| Inner and Outer Products Involving a Multivector | p. 44 |
| Inner and Outer Products between Higher Grades | p. 48 |
| Summary So Far | p. 50 |
| Exercises | p. 51 |
| (3+l)D Electromagnetics | p. 55 |
| The Lorentz Force | p. 55 |
| Maxwell's Equations in Free Space | p. 56 |
| Simplified Equations | p. 59 |
| The Connection between the Electric and Magnetic Fields | p. 60 |
| Plane Electromagnetic Waves | p. 64 |
| Charge Conservation | p. 68 |
| Multivector Potential | p. 69 |
| The Potential of a Moving Charge | p. 70 |
| Energy and Momentum | p. 76 |
| Maxwell's Equations in Polarizable Media | p. 78 |
| Boundary Conditions at an Interface | p. 84 |
| Exercises | p. 88 |
| Review of (3+l)D | p. 91 |
| Introducing Spacetime | p. 97 |
| Background and Key Concepts | p. 98 |
| Time as a Vector | p. 102 |
| The Spacetime Basis Elements | p. 104 |
| Spatial and Temporal Vectors | p. 106 |
| Basic Operations | p. 109 |
| Velocity | p. 111 |
| Different Basis Vectors and Frames | p. 112 |
| Events and Histories | p. 115 |
| Events | p. 115 |
| Histories | p. 115 |
| Straight-Line Histories and Their Time Vectors | p. 116 |
| Arbitrary Histories | p. 119 |
| The Spacetime Form of V | p. 121 |
| Working with Vector Differentiation | p. 123 |
| Working without Basis Vectors | p. 124 |
| Classification of Spacetime Vectors and Bivectors | p. 126 |
| Exercises | p. 127 |
| Relating Spacetime to (3+l)D | p. 129 |
| The Correspondence between the Elements | p. 129 |
| The Even Elements of Spacetime | p. 130 |
| The Odd Elements of Spacetime | p. 131 |
| From (3+l)D to Spacetime | p. 132 |
| Translations in General | p. 133 |
| Vectors | p. 133 |
| Bivectors | p. 135 |
| Trivectors | p. 136 |
| Introduction to Spacetime Splits | p. 137 |
| Some Important Spacetime Splits | p. 140 |
| Time | p. 140 |
| Velocity | p. 141 |
| Vector Derivatives | p. 142 |
| Vector Derivatives of General Multivectors | p. 144 |
| What Next? | p. 144 |
| Exercises | p. 145 |
| Change of Basis Vectors | p. 147 |
| Linear Transformations | p. 147 |
| Relationship to Geometric Algebras | p. 149 |
| Implementing Spatial Rotations and the Lorentz Transformation | p. 150 |
| Lorentz Transformation of the Basis Vectors | p. 153 |
| Lorentz Transformation of the Basis Bivectors | p. 155 |
| Transformation of the Unit Scalar and Pseudoscalar | p. 156 |
| Reverse Lorentz Transformation | p. 156 |
| The Lorentz Transformation with Vectors in Component Form | p. 158 |
| Transformation of a Vector versus a Transformation of Basis | p. 158 |
| Transformation of Basis for Any Given Vector | p. 162 |
| Dilations | p. 165 |
| Exercises | p. 166 |
| Further Spacetime Concepts | p. 169 |
| Review of Frames and Time Vectors | p. 169 |
| Frames in General | p. 171 |
| Maps and Grids | p. 173 |
| Proper Time | p. 175 |
| Proper Velocity | p. 176 |
| Relative Vectors and Paravectors | p. 178 |
| Geometric Interpretation of the Spacetime Split | p. 179 |
| Relative Basis Vectors | p. 183 |
| Evaluating Relative Vectors | p. 185 |
| Relative Vectors Involving Parameters | p. 188 |
| Transforming Relative Vectors and Paravectors to a Different Frame | p. 190 |
| Frame-Dependent versus Frame-Independent Scalars | p. 192 |
| Change of Basis for Any Object in Component Form | p. 194 |
| Velocity as Seen in Different Frames | p. 196 |
| Frame-Free Form of the Lorentz Transformation | p. 200 |
| Exercises | p. 202 |
| Application of the Spacetime Geometric Algebra to Basic Electromagnetics | p. 203 |
| The Vector Potential and Some Spacetime Splits | p. 204 |
| Maxwell's Equations in Spacetime Form | p. 208 |
| Maxwell's Free Space or Microscopic Equation | p. 208 |
| Maxwell's Equations in Polarizable Media | p. 210 |
| Charge Conservation and the Wave Equation | p. 212 |
| Plane Electromagnetic Waves | p. 213 |
| Transformation of the Electromagnetic Field | p. 217 |
| A General Spacetime Split for F | p. 217 |
| Maxwell's Equation in a Different Frame | p. 219 |
| Transformation of F by Replacement of Basis Elements | p. 221 |
| The Electromagnetic Field of a Plane Wave Under a Change of Frame | p. 223 |
| Lorentz Force | p. 224 |
| The Spacetime Approach to Electrodynamics | p. 227 |
| The Electromagnetic Field of a Moving Point Charge | p. 232 |
| General Spacetime Form of a Charge's Electromagnetic Potential | p. 232 |
| Electromagnetic Potential of a Point Charge in Uniform Motion | p. 234 |
| Electromagnetic Field of a Point Charge in Uniform Motion | p. 237 |
| Exercises | p. 240 |
| The Electromagnetic Field of a Point Charge Undergoing Acceleration | p. 243 |
| Working with Null Vectors | p. 243 |
| Finding F for a Moving Point Charge | p. 248 |
| Frad in the Charge's Rest Frame | p. 252 |
| Frad in the Observer's Rest Frame | p. 254 |
| Exercises | p. 258 |
| Conclusion | p. 259 |
| Appendices | p. 265 |
| Glossary | p. 265 |
| Axial versus True Vectors | p. 273 |
| Complex Numbers and the 2D Geometric Algebra | p. 274 |
| The Structure of Vector Spaces and Geometric Algebras | p. 275 |
| A Vector Space | p. 275 |
| A Geometric Algebra | p. 275 |
| Quaternions Compared | p. 281 |
| Evaluation of an Integral in Equation (5.14) | p. 283 |
| Formal Derivation of the Spacetime Vector Derivative | p. 284 |
| References | p. 287 |
| Further Reading | p. 291 |
| Index | p. 293 |
| Table of Contents provided by Ingram. All Rights Reserved. |
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