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### Geometric Algebra for Electrical Engineers

The multivector of geometric algebra extends vector algebra, providing a representation of points, directed line, plane, and volume segments.

Maxwell's equation in its multivector form is a compact and simple theoretical starting point for electromagnetic analysis.

This book details how geometric algebra may be applied to electromagnetic analysis and provides all of the prerequisite concepts and theory required to do so.

This book is suitable for students with an advanced undergraduate electrical engineering or physics background and aims to make the subject more accessible than the available research material.

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• Physics
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Email the Author(s) Peeter Joot

Peeter Joot is a math and physics enthusiast with a love of geometric algebra.

Peeter's education includes a B.A.Sc from UofT (1997 Engineering Science - computer engineering), 2019 UofT M.Eng (ECE electromagnetics), a lot of self-study, and non-degree study of most of the interesting 4th year UofT undergrad physics courses.

Peeter's day job is software development. He has over 20 years of experience with low level systems programing, operating system and hardware abstraction and exploitation, concurrency, and large scale refactoring. Peeter works for LzLabs and now has fun with PL/I, COBOL, JCL, VSAM, CICS and other legacy subsystems sure to scare away most developers of his generation.

Document Version

Dedication

Preface

Contents

List of Figures

1 Geometric Algebra.

1.1 Prerequisites.

1.1.1 Vector.

1.1.2 Vector space.

1.1.3 Basis, span and dimension.

1.1.4 Standard basis, length and normality.

1.2 Multivectors.

1.3 Colinear vectors.

1.4 Othogonal vectors.

1.5 Some nomenclature.

1.6 Two dimensions.

1.7 Plane rotations.

1.8 Duality.

1.9 Vector product, dot product and wedge product.

1.10 Reverse.

1.11 Complex representations.

1.12 Multivector dot product.

1.12.1 Dot product of a vector and bivector

1.12.2 Bivector dot product.

1.12.3 Problems.

1.13 Permutation within scalar selection.

1.14 Multivector wedge product.

1.14.1 Problems.

1.15 Projection and rejection.

1.16 Normal factorization of the wedge product.

1.17 The wedge product as an oriented area.

1.18 General rotation.

1.19 Symmetric and antisymmetric vector sums.

1.20 Reflection.

1.21 Linear systems.

1.22 A summary comparision.

1.23 Problem solutions.

2 Multivector calculus.

2.1 Reciprocal frames.

2.1.1 Motivation and definition.

2.1.2 R2 reciprocal frame.

2.1.3 R3 reciprocal frame.

2.1.4 Problems.

2.2 Curvilinear coordinates.

2.2.1 Two parameters.

2.2.2 Three (or more) parameters.

2.2.4 Vector derivative.

2.2.5 Examples.

2.2.6 Problems.

2.3 Integration theory.

2.3.1 Line integral.

2.3.2 Surface integral.

2.3.3 Volume integral.

2.3.4 Bidirectional derivative operators.

2.3.5 Fundamental theorem.

2.3.6 Stokes' theorem.

2.3.7 Fundamental theorem for Line integral.

2.3.8 Fundamental theorem for Surface integral.

2.3.9 Fundamental theorem for Volume integral.

2.4 Multivector Fourier transform and phasors.

2.5 Green's functions.

2.5.1 Motivation.

2.5.2 Green's function solutions.

2.5.3 Helmholtz equation.

2.5.4 First order Helmholtz equation.

2.6 Helmholtz theorem.

2.7 Problem solutions.

3 Electromagnetism.

3.1 Conventional formulation.

3.1.1 Problems.

3.2 Maxwell's equation.

3.3 Wave equation and continuity.

3.4 Plane waves.

3.5 Statics.

3.5.1 Inverting the Maxwell statics equation.

3.5.2 Enclosed charge.

3.5.3 Enclosed current.

3.5.4 Example field calculations.

3.6 Dynamics.

3.6.1 Inverting Maxwell's equation.

3.7 Energy and momentum.

3.7.1 Field energy and momentum density and the energy momentum tensor.

3.7.2 Poynting's theorem (prerequisites.)

3.7.3 Poynting theorem.

3.7.4 Examples: Some static fields.

3.7.5 Complex energy and power.

3.8 Lorentz force.

3.8.1 Statement.

3.8.2 Constant magnetic field.

3.9 Polarization.

3.9.1 Phasor representation.

3.9.2 Transverse plane pseudoscalar.

3.9.3 Pseudoscalar imaginary.

3.10 Transverse fields in a waveguide.

3.11 Multivector potential.

3.11.1 Definition.

3.11.2 Gauge transformations.

3.11.3 Far field.

3.12 Dielectric and magnetic media.

3.12.1 Statement.

3.12.2 Alternative form.

3.12.3 Gauge like transformations.

3.12.4 Boundary value conditions.

A Distribution theorems.

B Proof sketch for the fundamental theorem of geometric calculus.

C Green's functions.

C.1 Helmholtz operator.

C.2 Delta function derivatives.

D Energy momentum tensor for vector parameters.

Index

Bibliography

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