The book.
A draft of my book: Geometric Algebra for Electrical Engineers, is now available. I’ve supplied limited distribution copies of some of the early drafts and have had some good review comments of the chapter I (introduction to geometric algebra), and chapter II (multivector calculus) material, but none on the electromagnetism content. In defense of the reviewers, the initial version of the electromagnetism chapter, while it had a lot of raw content, was pretty exploratory and very rough. It’s been cleaned up significantly and is hopefully now more reader friendly.
Why I wrote this book.
I have been working on a part time M.Eng degree for a number of years. I wasn’t happy with the UofT ECE electromagnetics offerings in recent years, which have been inconsistently offered or unsatisfactory. For example: the microwave circuits course which sounded interesting, and had an interesting text book, was mind numbing, almost entirely about Smith charts. I had to go elsewhere to obtain the M.Eng degree requirements. That elsewhere was a project course.
I proposed a project to an electromagnetism project with the following goals
- Perform a literature review of applications of geometric algebra to the study of electromagnetism.
- Identify the subset of the literature that had direct relevance to electrical engineering.
- Create a complete, and as compact as possible, introduction to the prerequisites required for a graduate or advanced undergraduate electrical engineering student to be able to apply geometric algebra to problems in electromagnetism. With those prerequisites in place, work through the fundamentals of electromagnetism in a geometric algebra context.
In retrospect, doing this project was a mistake. I could have done this work outside of an academic context without paying so much (in both time and money). Somewhere along the way I lost track of the fact that I enrolled on the M.Eng to learn (it provided a way to take grad physics courses on a part time schedule), and got side tracked by degree requirements. Basically I fell victim to a “I may as well graduate” sentiment that would have been better to ignore. All that coupled with the fact that I did not actually get any feedback from my “supervisor”, who did not even read my work (at least so far after one year), made this project-course very frustrating. On the bright side, I really like what I produced, even if I had to do so in isolation.
Why geometric algebra?
Geometric algebra generalizes vectors, providing algebraic representations of not just directed line segments, but also points, plane segments, volumes, and higher degree geometric objects (hypervolumes.). The geometric algebra representation of planes, volumes and hypervolumes requires a vector dot product, a vector multiplication operation, and a generalized addition operation. The dot product provides the length of a vector and a test for whether or not any two vectors are perpendicular. The vector multiplication operation is used to construct directed plane segments (bivectors), and directed volumes (trivectors), which are built from the respective products of two or three mutually perpendicular vectors. The addition operation allows for sums of scalars, vectors, or any products of vectors. Such a sum is called a multivector.
The power to add scalars, vectors, and products of vectors can be exploited to simplify much of electromagnetism. In particular, Maxwell’s equations for isotropic media can be merged into a single multivector equation
\begin{equation}\label{eqn:quaternion2maxwellWithGA:20}
\lr{ \spacegrad + \inv{c} \PD{t}{}} \lr{ \BE + I c \BB } = \eta\lr{ c \rho – \BJ },
\end{equation}
where \( \spacegrad \) is the gradient, \( I = \Be_1 \Be_2 \Be_3 \) is the ordered product of the three R^3 basis vectors, \( c = 1/\sqrt{\mu\epsilon}\) is the group velocity of the medium, \( \eta = \sqrt{\mu/\epsilon} \), \( \BE, \BB \) are the electric and magnetic fields, and \( \rho \) and \( \BJ \) are the charge and current densities. This can be written as a single equation
\begin{equation}\label{eqn:ece2500report:40}
\lr{ \spacegrad + \inv{c} \PD{t}{}} F = J,
\end{equation}
where \( F = \BE + I c \BB \) is the combined (multivector) electromagnetic field, and \( J = \eta\lr{ c \rho – \BJ } \) is the multivector current.
Encountering Maxwell’s equation in its geometric algebra form leaves the student with more questions than answers. Yes, it is a compact representation, but so are the tensor and differential forms (or even the quaternionic) representations of Maxwell’s equations. The student needs to know how to work with the representation if it is to be useful. It should also be clear how to use the existing conventional mathematical tools of applied electromagnetism, or how to generalize those appropriately. Individually, there are answers available to many of the questions that are generated attempting to apply the theory, but they are scattered and in many cases not easily accessible.
Much of the geometric algebra literature for electrodynamics is presented with a relativistic bias, or assumes high levels of mathematical or physics sophistication. The aim of this work was an attempt to make the study of electromagnetism using geometric algebra more accessible, especially to other dumb engineering undergraduates like myself. In particular, this project explored non-relativistic applications of geometric algebra to electromagnetism. The end product of this project was a fairly small self contained book, titled “Geometric Algebra for Electrical Engineers”. This book includes an introduction to Euclidean geometric algebra focused on R^2 and R^3 (64 pages), an introduction to geometric calculus and multivector Green’s functions (64 pages), applications to electromagnetism (82 pages), and some appendices. Many of the fundamental results of electromagnetism are derived directly from the multivector Maxwell’s equation, in a streamlined and compact fashion. This includes some new results, and many of the existing non-relativistic results from the geometric algebra literature. As a conceptual bridge, the book includes many examples of how to extract familiar conventional results from simpler multivector representations. Also included in the book are some sample calculations exploiting unique capabilities that geometric algebra provides. In particular, vectors in a plane may be manipulated much like complex numbers, which has a number of advantages over working with coordinates explicitly.
Followup.
In many ways this work only scratches the surface. Many more worked examples, problems, figures and computer algebra listings should be added. In depth applications of derived geometric algebra relationships to problems customarily tackled with separate electric and magnetic field equations should also be incorporated. There are also theoretical holes, topics covered in any conventional introductory electromagnetism text, that are missing. Examples include the Fresnel relationships for transmission and reflection at an interface, in depth treatment of waveguides, dipole radiation and motion of charged particles, bound charges, and meta materials to name a few. Many of these topics can probably be handled in a coordinate free fashion using geometric algebra. Despite all the work that is required to help bridge the gap between formalism and application, making applied electromagnetism using geometric algebra truly accessible, it is my belief this book makes some good first steps down this path.
The choice that I made to completely avoid the geometric algebra space-time-algebra (STA) is somewhat unfortunate. It is exceedingly elegant, especially in a relativisitic context. Despite that, I think that this was still a good choice from a pedagogical point of view, as most of the prerequisites for an STA based study will have been taken care of as a side effect, making that study much more accessible.
I find this very interesting. I have very little knowledge of GA. What is its advantages over differential forms?
This is not an answer, but a question: Given the differential forms representation of Maxwell’s equation:
\begin{equation}
\begin{aligned}
d \alpha &= dj \\
d *\alpha &= 0,
\end{aligned}
\end{equation}
is it possible to directly invert Maxwell’s equations to find \(\alpha\) given a current \(j\)? That’s possible with geometric algebra, as we can find the Green’s function for \(\spacegrad + (1/c)\partial_t\) (or the frequency domain equivalent).
In exploring geometric algebra applications there were many instances where I was not able to use Stokes’ theorem (for which there is an equivalent of the general Stokes’ theorem from differential forms). Instead I had to use (or it was more natural to use) more general multivector integrals evaluated with the “Fundamental theorem of geometric calculus”.
I personally find the two form for the field:
\begin{equation}\label{eqn:ece2500report:180}
\alpha = \lr{ E_1 dx^1 + E_2 dx^2 + E_3 dx^3 }(c dt) + H_1 dx^2 dx^3 + H_2 dx^3 dx^1 + H_3 dx^1 dx^2.
\end{equation}
rather unintuitive. Having to use \(c dx^i t\) as a direction vector isn’t exactly obvious. There are a number of operations that I can do with conventional vector algebra that I’m not sure how to represent using differential forms. I am not an expert (or even novice) with differential forms, but from what I have seen, it is easier to make the transition from conventional vector algebra to geometric algebra (or visca versa) than doing the same with differential forms. With geometric algebra, we get the power of differential forms for manifold calculus and curvilinear coordinates when we want it, but don’t have the cost of having to use forms when plain old vector basis vectors do just as nicely.
I’ve been thinking of diving into geometric algebra for a while, one thing I’ve been wondering though is how applicable the theory is. Suppose I had a good understanding of geometric algebra, would it be possible for example for me to go through Jackson and solve most of the problems there in terms of this formalism? Would it be any easier than doing things the ‘usual way’?
Most of what is in my book so far is the basic context required to attempt the problems you’ve indicated. I wanted to attempt a number of such problems too, but have only scratched the surface in that respect. My suspicion is that the applicability of GA will be problem specific, but that it will be an excellent tool for some problems.
I did work one of the transmission line derivations from Jackson (transmission line relations between the transverse and propagation direction field components) in this book using GA and both the result and the method are pretty elegant.
Thanks for that, that does sound interesting. I think I will take a look.
One other question, while searching around I’ve also heard Kähler calculus mentioned as something different to geometric calculus, notably in https://altermanschool2017.uacg.bg/comparisonKC-GC.pdf where the author claims it to be superior, though without any concrete details. Have you by any chance come across this before, or know how it relates to geometric calculus?
No, I’m not familiar with Kähler calculus at this point. With respect to electromagnetism, the context of manifold calculus (curvilinear coordinates, reciprocal frames, tangent spaces, …) is useful to derive various theorems (fundamental theorem of geometric calculus, generalized Stokes’ theorem, …), but after that we are able to use those and no longer care about the manifold calculus ideas. Given that, I suspect that Kähler calculus won’t have much utility for GA+E&M.