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El. knyga: Understanding Geometric Algebra for Electromagnetic Theory

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This book aims to disseminate geometric algebra as a straightforward mathematical tool set for working with and understanding classical electromagnetic theory. It's target readership is anyone who has some knowledge of electromagnetic theory, predominantly ordinary scientists and engineers who use it in the course of their work, or postgraduate students and senior undergraduates who are seeking to broaden their knowledge and increase their understanding of the subject. It is assumed that the reader is not a mathematical specialist and is neither familiar with geometric algebra or its application to electromagnetic theory. The modern approach, geometric algebra, is the mathematical tool set we should all have started out with and once the reader has a grasp of the subject, he or she cannot fail to realize that traditional vector analysis is really awkward and even misleading by comparison. Professors can request a solutions manual by email: pressbooks@ieee.org

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"This book will benefit scientists and engineers who use electromagnetic theory in the course of their work.  (Zentralblatt MATH, 1 May 2013)

Preface xi
Reading Guide xv
1 Introduction
1(6)
2 A Quick Tour of Geometric Algebra
7(20)
2.1 The Basic Rules of a Geometric Algebra
16(1)
2.2 3D Geometric Algebra
17(2)
2.3 Developing the Rules
19(5)
2.3.1 General Rules
20(1)
2.3.2 3D
21(1)
2.3.3 The Geometric Interpretation of Inner and Outer Products
22(2)
2.4 Comparison with Traditional 3D Tools
24(1)
2.5 New Possibilities
24(2)
2.6 Exercises
26(1)
3 Applying the Abstraction
27(12)
3.1 Space and Time
27(1)
3.2 Electromagnetics
28(4)
3.2.1 The Electromagnetic Field
28(2)
3.2.2 Electric and Magnetic Dipoles
30(2)
3.3 The Vector Derivative
32(2)
3.4 The Integral Equations
34(2)
3.5 The Role of the Dual
36(1)
3.6 Exercises
37(2)
4 Generalization
39(16)
4.1 Homogeneous and Inhomogeneous Multivectors
40(1)
4.2 Blades
40(2)
4.3 Reversal
42(1)
4.4 Maximum Grade
43(1)
4.5 Inner and Outer Products Involving a Multivector
44(4)
4.6 Inner and Outer Products between Higher Grades
48(2)
4.7 Summary So Far
50(1)
4.8 Exercises
51(4)
5 (3+1)D Electromagnetics
55(36)
5.1 The Lorentz Force
55(1)
5.2 Maxwell's Equations in Free Space
56(3)
5.3 Simplified Equations
59(1)
5.4 The Connection between the Electric and Magnetic Fields
60(4)
5.5 Plane Electromagnetic Waves
64(4)
5.6 Charge Conservation
68(1)
5.7 Multivector Potential
69(7)
5.7.1 The Potential of a Moving Charge
70(6)
5.8 Energy and Momentum
76(2)
5.9 Maxwell's Equations in Polarizable Media
78(10)
5.9.1 Boundary Conditions at an Interface
84(4)
5.10 Exercises
88(3)
6 Review of (3+1)D
91(6)
7 Introducing Space time
97(32)
7.1 Background and Key Concepts
98(4)
7.2 Time as a Vector
102(2)
7.3 The Spacetime Basis Elements
104(5)
7.3.1 Spatial and Temporal Vectors
106(3)
7.4 Basic Operations
109(2)
7.5 Velocity
111(1)
7.6 Different Basis Vectors and Frames
112(3)
7.7 Events and Histories
115(6)
7.7.1 Events
115(1)
7.7.2 Histories
115(1)
7.7.3 Straight-Line Histories and Their Time Vectors
116(3)
7.7.4 Arbitrary Histories
119(2)
7.8 The Spacetime Form of δ
121(2)
7.9 Working with Vector Differentiation
123(1)
7.10 Working without Basis Vectors
124(2)
7.11 Classification of Spacetime Vectors and Bivectors
126(1)
7.12 Exercises
127(2)
8 Relating Spacetime to (3+1)D
129(18)
8.1 The Correspondence between the Elements
129(4)
8.1.1 The Even Elements of Spacetime
130(1)
8.1.2 The Odd Elements of Spacetime
131(1)
8.1.3 From (3+1)D to Spacetime
132(1)
8.2 Translations in General
133(4)
8.2.1 Vectors
133(2)
8.2.2 Biveetors
135(1)
8.2.3 Trivectors
136(1)
8.3 Introduction to Spacetime Splits
137(3)
8.4 Some Important Spacetime Splits
140(4)
8.4.1 Time
140(1)
8.4.2 Velocity
141(1)
8.4.3 Vector Derivatives
142(2)
8.4.4 Vector Derivatives of General Multivectors
144(1)
8.5 What Next?
144(1)
8.6 Exercises
145(2)
9 Change of Basis Vectors
147(22)
9.1 Linear Transformations
147(2)
9.2 Relationship to Geometric Algebras
149(1)
9.3 Implementing Spatial Rotations and the Lorentz Transformation
150(3)
9.4 Lorentz Transformation of the Basis Vectors
153(2)
9.5 Lorentz Transformation of the Basis Bivectors
155(1)
9.6 Transformation of the Unit Scalar and Pseudoscalar
156(1)
9.7 Reverse Lorentz Transformation
156(2)
9.8 The Lorentz Transformation with Vectors in Component Form
158(7)
9.8.1 Transformation of a Vector versus a Transformation of Basis
158(4)
9.8.2 Transformation of Basis for Any Given Vector
162(3)
9.9 Dilations
165(1)
9.10 Exercises
166(3)
10 Further Spacetime Concepts
169(34)
10.1 Review of Frames and Time Vectors
169(2)
10.2 Frames in General
171(2)
10.3 Maps and Grids
173(2)
10.4 Proper Time
175(1)
10.5 Proper Velocity
176(2)
10.6 Relative Vectors and Paravectors
178(14)
10.6.1 Geometric Interpretation of the Spacetime Split
179(4)
10.6.2 Relative Basis Vectors
183(2)
10.6.3 Evaluating Relative Vectors
185(3)
10.6.4 Relative Vectors Involving Parameters
188(2)
10.6.5 Transforming Relative Vectors and Paravectors to a Different Frame
190(2)
10.7 Frame-Dependent versus Frame-Independent Scalars
192(2)
10.8 Change of Basis for Any Object in Component Form
194(2)
10.9 Velocity as Seen in Different Frames
196(4)
10.10 Frame-Free Form of the Lorentz Transformation
200(2)
10.11 Exercises
202(1)
11 Application of the Spacetime Geometric Algebra to Basic Electromagnetics
203(40)
11.1 The Vector Potential and Some Spacetime Splits
204(4)
11.2 Maxwell's Equations in Spacetime Form
208(4)
11.2.1 Maxwell's Free Space or Microscopic Equation
208(2)
11.2.2 Maxwell's Equations in Polarizable Media
210(2)
11.3 Charge Conservation and the Wave Equation
212(1)
11.4 Plane Electromagnetic Waves
213(4)
11.5 Transformation of the Electromagnetic Field
217(7)
11.5.1 A General Spacetime Split for F
217(2)
11.5.2 Maxwell's Equation in a Different Frame
219(2)
11.5.3 Transformation off by Replacement of Basis Elements
221(2)
11.5.4 The Electromagnetic Field of a Plane Wave Under a Change of Frame
223(1)
11.6 Lorentz Force
224(3)
11.7 The Spacetime Approach to Electrodynamics
227(5)
11.8 The Electromagnetic Field of a Moving Point Charge
232(8)
11.8.1 General Spacetime Form of a Charge's Electromagnetic Potential
232(2)
11.8.2 Electromagnetic Potential of a Point Charge in Uniform Motion
234(3)
11.8.3 Electromagnetic Field of a Point Charge in Uniform Motion
237(3)
11.9 Exercises
240(3)
12 The Electromagnetic Field of a Point Charge Undergoing Acceleration
243(16)
12.1 Working with Null Vectors
243(5)
12.2 Finding F for a Moving Point Charge
248(4)
12.3 Frad in the Charge's Rest Frame
252(2)
12.4 Frad in the Observer's Rest Frame
254(4)
12.5 Exercises
258(1)
13 Conclusion
259(6)
14 Appendices
265(22)
14.1 Glossary
265(8)
14.2 Axial versus True Vectors
273(1)
14.3 Complex Numbers and the 2D Geometric Algebra
274(1)
14.4 The Structure of Vector Spaces and Geometric Algebras
275(6)
14.4.1 A Vector Space
275(1)
14.4.2 A Geometric Algebra
275(6)
14.5 Quaternions Compared
281(2)
14.6 Evaluation of an Integral in Equation (5.14)
283(1)
14.7 Formal Derivation of the Spacetime Vector Derivative
284(3)
References 287(4)
Further Reading 291(2)
Index 293
JOHN W. ARTHUR earned his PhD from Edinburgh University in 1974 for research into light scattering in crystals. He has been involved in academic research, the microelectronics industry, and corporate R&D. Dr. Arthur has published various research papers in acclaimed journals, including IEEE Antennas and Propagation Magazine. His 2008 paper entitled "The Fundamentals of Electromagnetic Theory Revisited" received the 2010 IEEE Donald G. Fink Prize for Best Tutorial Paper. A senior member of the IEEE, Dr. Arthur was elected a fellow of the Royal Society of Edinburgh and of the United Kingdom's Royal Academy of Engineering in 2002. He is currently an honorary fellow in the School of Engineering at the University of Edinburgh.
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