# Geometric algebra for Electrical Engineers

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### Peeter Joot

Jan 18, 2018, 10:03:34 PM1/18/18
to Geometric_Algebra
I've been attempting to write a "for engineers" treatment of the electromagnetism using Geometric Algebra.

I know that there are a number of existing books and papers with significant GA+electromagnetism content (in particular: Doran & Lasenby's "GA for Physicists", Baylis's "Electrodynamics, a modern geometric approach", and Jancewicz's "Multivectors and clifford algebra in electrodynamics").  I've found that a lot of the GA+E&M literature has a strong relativistic bias (which is arguably natural), or are just generally difficult (ie. GAFP is an awesome book, but a lot of the material requires years of prerequisite study to understand).  Requiring both relativity, and GA makes the material fairly inaccessible, as a double learning curve must be hurdled.

What I've written avoids any relativity (at least so far), and I've been writing with a target audience of upper level undergrad electrical engineers or early graduate level.  I'm not sure how successful I've been "dumbing" the subject down to the level of an electrical engineer (no insult intended, as I'm one of them), and I'd be interested in getting some feedback.

peeterjoot AT protonmail DOT com

You'll find some new material (that I haven't seen in papers anywhere), as well as a number of the existing applications that can be found in the literature.

Outline:

part.1   Geometric Algebra.
chapter.1   Geometric Algebra.
section.1.1 Prerequisites.
subsection.1.1.1  Vector space.
subsection.1.1.2  Basis, span and dimension.
subsection.1.1.3  Standard basis, length and normality.
section.1.2 Definitions
subsection.1.2.1  Multivector space.
subsection.1.2.2  Nomenclature.
section.1.3 Analysis.
subsection.1.3.1  Colinear vectors.
subsection.1.3.2  Normal vectors.
subsection.1.3.3  2D multiplication table.
subsection.1.3.4  Plane rotations.
subsection.1.3.5  Vector product, dot product and wedge product.
subsection.1.3.6  Reverse.
subsection.1.3.7  Complex representations.
subsection.1.3.8  Multivector dot product.
subsection.1.3.9  Permutation within scalar selection.
subsection.1.3.10 Multivector wedge product.
subsection.1.3.11 Duality.
subsection.1.3.12 Projection and rejection.
subsection.1.3.13 Normal factorization of the wedge product.
subsection.1.3.14 The wedge product as an oriented area.
subsection.1.3.15 General rotation.
subsection.1.3.16 Symmetric and antisymmetric vector sums.
subsection.1.3.17 Reflection.
subsection.1.3.18 Linear systems.
section.1.4 A summary comparision.
section.1.5 Problem solutions.
chapter.2   Multivector calculus.
section.2.1 Reciprocal frames.
subsection.2.1.1  Problems.
section.2.2 Curvilinear coordinates.
subsection.2.2.1  Cylindrical coordinates.
subsection.2.2.2  Spherical coordinates.
subsection.2.2.3  Toroidal coordinates.
subsection.2.2.4  Problems.
section.2.3 Green's theorem.
section.2.4 Stokes' theorem.
subsection.2.4.1  Statement.
subsection.2.4.2  One parameter specialization of Stokes' theorem.
subsection.2.4.3  Two parameter specialization of Stokes' theorem.
subsection.2.4.4  Three parameter specialization of Stokes' theorem.
subsection.2.4.5  Using scalar volume elements
subsection.2.4.6  Problems
section.2.5 Fundamental theorem of geometric calculus.
subsection.2.5.1  Fundamental Theorem of Geometric Calculus.
subsection.2.5.2  Green's function for the gradient in Euclidean spaces.
subsection.2.5.3  Helmholtz theorem.
section.2.6 Problem solutions.
part.2}{Electromagnetism.
chapter.3   Electromagnetism.
section.3.1 Maxwell and Lorentz equations.
subsection.3.1.1  Problems.
section.3.2 Electrostatics.
subsection.3.2.1  Enclosed charge.
subsection.3.2.2  Electric potential.
subsection.3.2.4  Poisson equation solution.
subsection.3.2.5  Example: Straight line charge.
subsection.3.2.6  Example: Circular line charge.
subsection.3.2.7  Problems.
section.3.3 Magnetostatics.
subsection.3.3.1  Vector potential.
subsection.3.3.2  Enclosed current density.
subsection.3.3.3  Enclosed current.
subsection.3.3.4  Biot-Savart law.
subsection.3.3.5  Example. Ampere's law for magnetic field between two current sources.
section.3.4 Maxwell's equation GA.
subsection.3.4.1  Wave equation.
subsection.3.4.2  Continuity equations.
section.3.5 Statics.
subsection.3.5.1  Inverting the Maxwell statics equation.
subsection.3.5.2  Example. Infinite line charge and current.
subsection.3.5.3  Example. Infinite planar charge and current.
subsection.3.5.4  Example. Field of a ring of charge or current density.
section.3.6 Energy and Momentum.
subsection.3.6.1  Field energy and momentum density and the stress energy tensor.
subsection.3.6.2  Poynting's theorem.
subsection.3.6.3  Relation to Lorentz force.
subsection.3.6.4  Example. Energy density and Poynting vectors for static field solutions.
subsection.3.6.5  Complex energy and power.
section.3.7 Plane waves.
section.3.8 Polarization.
subsection.3.8.1  Plane wave.
subsection.3.8.2  Circular polarization basis.
subsection.3.8.3  Linear polarization.
subsection.3.8.4  Other phase dependence and energy momentum.
subsection.3.8.5  Elliptical parameterization.
subsection.3.8.6  Pseudoscalar imaginary.
subsection.3.8.7  Problems.
section.3.9 Transverse fields in a waveguide.
section.3.10   Boundary value conditions.
section.3.11   Multivector potential.
subsection.3.11.1 General potential representation.
subsection.3.11.2 Electric sources.
subsection.3.11.3 Magnetic sources.
subsection.3.11.4 Far field.
subsection.3.11.5 Gauge transformations
subsection.3.11.6 Lorenz gauge
section.3.12   Lorentz force.
subsection.3.12.1 GA statement.
subsection.3.12.2 Constant magnetic field.
section.3.13   Dielectric and magnetic media.
part.3   Backmatter
appendix.A  Justifying the contraction axiom.
appendix.B  Distribution theorems.
appendix.C  GA electrodynamics in the literature.
appendix.C  Index
dummy.4  Bibliography

### Allan Cortzen

Jan 19, 2018, 3:31:09 AM1/19/18
I am interested.
Allan

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### Peeter Joot

Apr 5, 2018, 7:16:39 PM4/5/18
to Geometric_Algebra
I've posted a public draft of my book.  A link is available here:

http://peeterjoot.com/2018/04/05/i-wrote-a-book-geometric-algebra-for-electrical-engineers/

I had some good feedback on chapter I & II (intro to GA and multivector calculus), and those chapters have been significantly improved in response to that feedback.  The electromagnetism content in the original drafts that I'd made available was very rough and exploratory.  It has now been reworked and is much more systematic now.  In my opinion, that chapter is now significantly better, and will hopefully no longer scare off prospective readers.

I'd appreciate any feedback anybody is willing to offer, especially for the electromagnetism chapter (3).

Peeter