Geometric Algebra for Electrical Engineers Multivector electromagnetism

Description

This book introduces the fundamentals of geometric algebra and calculus, and applies those tools to the study of electromagnetism. Geometric algebra provides a structure that can represent oriented point, line, plane, and volume segments. Vectors, which can be thought of as a representation of oriented line segments, are generalized to multivectors. A full fledged, but non-commutative (i.e. order matters) multiplication operation will be defined for products of vectors. Namely, the square of a vector is the square of its length. This simple rule, along with a requirement that we can sum vectors and their products, essentially defines geometric algebra. Such sums of scalars, vectors and vector products are called multivectors. The reader will see that familiar concepts such as the dot and cross product are related to a more general vector product, and that algebraic structures such as complex numbers can be represented as multivectors. We will be able to utilize generalized complex exponentials to do rotations in arbitrarily oriented planes in space, and will find that simple geometric algebra representations of many geometric transformations are possible. Generalizations of the divergence and Stokes’ theorems are required once we choose to work with multivector functions. There is an unfortunate learning curve required to express this generalization, but once overcome, we will be left with a single powerful multivector integration theorem that has no analogue in conventional vector calculus. This fundamental theorem of geometric calculus incorporates Green’s (area) theorem, the divergence theorem, Stokes’ theorems, and complex residue calculus. Multivector calculus also provides the opportunity to define a few unique and powerful Green’s functions that almost trivialize solutions of Maxwell’s equations. Instead of working separately with electric and magnetic fields, we will work with a hybrid multivector field that includes both electric and magnetic field contributions, and with a multivector current that includes both charge and current densities. The natural representation of Maxwell’s equations is a single multivector equation that is easier to solve and manipulate then the conventional mess of divergence and curl equations are familiar to the reader. This book is aimed at graduate or advanced undergraduates in electrical engineering or physics. While all the fundamental results of electromagnetism are derived from Maxwell’s equations, there will be no attempt to motivate Maxwell’s equations themselves, so existing familiarity with the subject is desirable. uploaded by zhang. Copyright 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 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. 2.2.3 Gradient. 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. 2.5.5 Spacetime gradient. 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. E Differential forms vs geometric calculus. Index Bibliography