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Peeter Joot’s Weblog » Geometric Algebra for Electrical Engineers.

Peeter Joot’s Weblog » Geometric Algebra for Electrical Engineers.

2023-09-24 03:52:38

My first guide, Geometric Algebra for Electrical Engineers is accessible within the following codecs.

The softcover and hardcover variations are each printed utilizing Amazon’s kindle direct publishing and have a 6×9″ format (318 pages.)  I used to even have a colour print model out there, however unpublished that when the hardcover choice grew to become out there (however the free pdf model is colour.)  If you’re not within the USA, search for the guide in your native Amazon market, because it has been printed with expanded distribution.

Be happy to e-mail me with suggestions, questions or feedback. I’ve additionally allotted a discord server for myself with the latex bot running, so if in case you have questions that require backwards and forwards, or math formatted chat, that’s superior to e-mail, twitter DMs, linked-in messages, or varied different ways in which I’m out there.

In case you obtain the free PDF and really feel undercharged, feel free to send some bitcoin my way.


Why I wrote this guide.

This guide was the product of an M.Eng “undertaking course”, the place my intention was to

  1. Carry out a literature evaluation of purposes of geometric algebra to the examine of electromagnetism.
  2. Establish the subset of the literature that had direct relevance to electrical engineering.
  3. Create a whole, and as compact as attainable, introduction to the conditions required for a graduate or superior undergraduate electrical engineering scholar to have the ability to apply geometric algebra to issues in electromagnetism. With these conditions in place, work via the basics of electromagnetism in a geometrical algebra context.
  4. Optionally, time allowing, evaluation current and/or create some Laptop Algebra Software program (CAS) for symbolic Geometric Algebra.

This served to supply the engineering credit I wanted to graduate, but additionally was a chance to assemble a few of my examine associated to Geometric Algebra right into a coherent narrative.  That is in stark distinction to my “Exploring Physics with Geometric Algebra” writing, which was assembled from many particular person articles, and is stuffed with redundancy, and was written for myself on the level of writing.

Why geometric algebra?

Geometric algebra generalizes vectors, offering algebraic representations of not simply directed line segments, but additionally factors, airplane segments, volumes, and better diploma geometric objects (hypervolumes.). The geometric algebra illustration of planes, volumes and hypervolumes requires a vector dot product, a vector multiplication operation, and a generalized addition operation. The dot product gives the size of a vector and a check for whether or not or not any two vectors are perpendicular. The vector multiplication operation is used to assemble directed airplane segments (bivectors), and directed volumes (trivectors), that are constructed from the respective merchandise of two or three mutually perpendicular vectors. The addition operation permits for sums of scalars, vectors, or any merchandise of vectors. Such a sum is named a multivector.

The facility so as to add scalars, vectors, and merchandise of vectors will be exploited to simplify a lot of electromagnetism. Specifically, Maxwell’s equations for isotropic media will be merged right into a single multivector equation
lr{ spacegrad + inv{c} PD{t}{}} lr{ BE + I c BB } = etalr{ c rho – BJ },
the place ( spacegrad ) is the gradient, ( I = Be_1 Be_2 Be_3 ) is the ordered product of the three R^3 foundation vectors, ( c = 1/sqrt{muepsilon}) is the group velocity of the medium, ( eta = sqrt{mu/epsilon} ), ( BE, BB ) are the electrical and magnetic fields, and ( rho ) and ( BJ ) are the cost and present densities. This may be written as a single equation
lr{ spacegrad + inv{c} PD{t}{}} F = J,
the place ( F = BE + I c BB ) is the mixed (multivector) electromagnetic discipline, and ( J = etalr{ c rho – BJ } ) is the multivector present.

Encountering Maxwell’s equation in its geometric algebra kind leaves the coed with extra questions than solutions. Sure, it’s a compact illustration, however so are the tensor and differential varieties (and even the quaternionic) representations of Maxwell’s equations. The coed must know learn how to work with the illustration whether it is to be helpful. It must also be clear learn how to use the prevailing typical mathematical instruments of utilized electromagnetism, or learn how to generalize these appropriately. Individually, there are solutions out there to lots of the questions which are generated trying to use the idea, however they’re scattered and in lots of circumstances not simply accessible.

A lot of the geometric algebra literature for electrodynamics is introduced with a relativistic bias, or assumes excessive ranges of mathematical or physics sophistication. The intention of this work was an try and make the examine of electromagnetism utilizing geometric algebra extra accessible, particularly to different dumb engineering undergraduates like myself. Specifically, this undertaking explored non-relativistic purposes of geometric algebra to electromagnetism. The top product of this undertaking was a reasonably small self contained guide, titled “Geometric Algebra for Electrical Engineers”. This guide contains an introduction to Euclidean geometric algebra centered on R^2 and R^3 (64 pages), an introduction to geometric calculus and multivector Inexperienced’s features (64 pages), purposes to electromagnetism (82 pages), and a few appendices. Most of the elementary outcomes of electromagnetism are derived instantly from the multivector Maxwell’s equation, in a streamlined and compact vogue. This contains some new outcomes, and lots of the current non-relativistic outcomes from the geometric algebra literature. As a conceptual bridge, the guide contains many examples of learn how to extract acquainted typical outcomes from easier multivector representations. Additionally included within the guide are some pattern calculations exploiting distinctive capabilities that geometric algebra gives. Specifically, vectors in a airplane could also be manipulated very like advanced numbers, which has an a variety of benefits over working with coordinates explicitly.


In some ways this work solely scratches the floor. Many extra labored examples, issues, figures and pc algebra listings must be added. In depth purposes of derived geometric algebra relationships to issues usually tackled with separate electrical and magnetic discipline equations must also be integrated. There are additionally theoretical holes, subjects coated in any typical introductory electromagnetism textual content, which are lacking. Examples embrace the Fresnel relationships for transmission and reflection at an interface, in depth remedy of waveguides, dipole radiation and movement of charged particles, sure prices, and meta supplies to call just a few. Many of those subjects can in all probability be dealt with in a coordinate free vogue utilizing geometric algebra. Regardless of all of the work that’s required to assist bridge the hole between formalism and utility, making utilized electromagnetism utilizing geometric algebra really accessible, it’s my perception this guide makes some good first steps down this path.

The selection that I made to utterly keep away from the geometric algebra space-time-algebra (STA) is considerably unlucky. It’s exceedingly elegant, particularly in a relativisitic context. Regardless of that, I believe that this was nonetheless a sensible choice from a pedagogical standpoint, as many of the conditions for an STA primarily based examine could have been taken care of as a facet impact, making that examine far more accessible.

Mathematica packages for Geometric Algebra.

I initially had some bother with among the current Mathematica packages that I discovered for Geometric Algebra (i.e. they might grasp my Mathematica front-end intermittently), and ended up writing a pair much less fancy Mathematica packages myself.  These will be present in my repository

There are 4 Mathematica packages, Cl20.m, GA20.m, GA30.m and GA13.m.

  • Cl20.m makes use of two advanced numbers to characterize the (2D Euclidean) algebra.
  • GA20.m and GA30.m use Pauli matrices to characterize the respective (2D and 3D) algebras
  • GA13.m makes use of makes use of Dirac matrices as a illustration of the area time algebra

Every of those packages use much less difficult Mathematica syntax than lots of the current packages that I discovered (and didn’t grasp my Mathematica entrance finish).  For the examples within the guide I ended up utilizing an current (and doubtless extra well-known) package deal, CliffordBasic.m geometric algebra module, as a substitute of my much less common (and maybe extra hacky) implementation.

Suggestions or contribution.

Must you want to actively contribute typo fixes (or much more important adjustments) to this guide, you are able to do so by contacting me, or by forking your personal copy of the related git repositories and constructing the guide pdf from supply, and submitting a subsequent merge request.


git clone [email protected]:peeterjoot/latex-notes-compilations.git peeterjoot
cd peeterjoot

submods="figures/GAelectrodynamics figures/gabook mathematica GAelectrodynamics gapauli latex frequencydomain"
for i in $submods ; do
   git submodule replace --init $i
   (cd $i && git checkout grasp)
export PATH=`pwd`/latex/bin:$PATH

cd GAelectrodynamics
make mmacells.sty all

I reserve the fitting to impose dictatorial management over any enhancing and content material selections, and should not settle for merge requests as-is, or in any respect. That stated, I’ll in all probability not refuse cheap solutions or merge requests.

Experimenting with pricing of the guide.

In mid Could 2020, I raised the worth of the paperback model barely (from $12 to $14.50 USD).  That’s a 17% worth improve, however the worth continues to be fairly low from an absolute worth perspective.  This worth improve was an experiment in response to a reseller (SuperBookDeals) shopping for copies at $12 after which reselling them at increased costs.  For some cause amazon lists the higher price reseller copies before their own direct sales version, so a purchaser needed to exit of their solution to discover the bottom priced model.  I wouldn’t care if resellers undercut the checklist worth, after which bought a preferential itemizing from amazon, however the truth that they do that at the next worth suggests I’d set the worth too low.

In Aug 2023, I raised the worth additional, reflecting Amazon printing value will increase.

If you’re considering a replica of the guide, however don’t like the worth, see above for a hyperlink to the present (colour) PDF model, which continues to be out there without cost.


  • V0.1.19-2 (Sep 2, 2023)
    • Reworked lots of the Mathematica generated figures.  Now utilizing the MaTeX[] extension to do the determine labelling, because it appears to be like a lot better, and is per the fonts within the textual content.

      Every of those are individually very small adjustments, barely noticeable, however I believe it makes a pleasant distinction to general high quality.

      See Also

      In lots of circumstances, I’ve generated new separate figures for the black and white amazon paper editions of the guide, utilizing straight black as a substitute of colours, in order that they don’t look as washed out.

  • V0.1.18-4 (Aug 22, 2023)
    • Inner: try to make use of tex4ebook to create an epub/mobi format that can work as a kindle guide.  This failed horribly, as a consequence of kindle’s processing step (additionally seen with Kindle previewer 3), which rescales all of the equation photos to make them into too-big squares (different standalone viewers like Calibre don’t mangle the pictures this fashion.)

    • Rework the spherical unit vector illustration instance, to make use of ( mathbf{e}_3 e^{j(theta + pi/2)} ) for the polar unit vector, which is extra compact and easier to control.
    • Inner: specialize the colours used for the uploads (i.e.: the paperback and hardcover variations on amazon) to make use of Black and White as a substitute of the DarkOliveGreen/Maroon colour scheme that I exploit for the web model.  This could makes among the textual content present up much less gray within the printed model of the guide.
  • V0.1.18-3 (Aug 14, 2023)
  • V0.1.18-2 (Aug 1, 2023)
    • Nicholas: 2.3.1 first sentance: typo: this could say “…incorporates a product…” and may finish with a interval.

    • Nicholas: typo: Within the textual content above determine 2.1, (4,2)) has one too many parentheses.

    • Nicholas: Part 2.2 is entitled “Curvilinear Coordinates”.  And definition 2.2 is entitled the identical.  However curvilinear coordinates should not explicitly outlined; as a substitute, the curvilinear foundation is outlined.  It’s fairly apparent that the coordinates are these with respect to the premise.  However maybe you’d both like to vary the title to “Curvilinear foundation” or explicitly outline what the coordinates are?

    • 2.98b isn’t apparent.  Add that as an issue.

  • V0.1.18-1 (Jul 30, 2023)

    • Nicholas:
      •    In (2.86), I’m undecided what hat{ rho } and hat{ phi } are.  How would I do know that hat{ phi } = (1/rho) x_phi ?  — Clarified.
      •    Typo. grad rho = ( cos phi, sin phi ) in equation (2.84).  That’s, I believe the 1 / rho shouldn’t be within the first line of that equation.
      •    Typo: x_phi within the paragraph following (2.75b) ought to have a rho in it: x_phi = rho e_2 exp( i phi ).
      •    Typo: to be per the remainder of the guide, I believe that the final expression of (2.72) must be a delta with a subscript and a superscript, however with out a area within the subscript.
    • surfaceintegral.tex   

  • V0.1.17-11 (Jul 16, 2023)
  • V0.1.17-9 (Jul 14, 2023)
    • Nicholas Dwark: The “of” after “subspace” shouldn’t be there.  For our the operate equation (2.31) our …” must be “For equation (2.31), our …”
  • V0.1.17-8 (Jul 4, 2023)
    • Nicholas Dwark: triple product -> the triple product.  Use scalar triple product and vector triple product the place acceptable.
    • Within the sentence after equation (1.107), “part” is singular, so “lie” must be “lies”.

    • (1.103) is described as a bivector-trivector dot product.  However I believe that e1+2e2 is a vector, and it must be a “vector-trivector” dot product.

  • V0.1.17-7 (Jul 3, 2023)
    • Nicholas Dwark: “theorem 1.3” is mistakenly repeated twice in footnote 3 below determine 1.7 in your guide.
  • V0.1.17-5 (Mar 26, 2023)
    • Run preface.tex via spellcheck like every part else. Due to Pippy on discord for stating the “dimemsion” spelling error.
    • Purge remaining dmath’s
    • reply for downside:isotropicMaxwells:720
    • isotropicMaxwells.tex: reply for downside:isotropicMaxwells:700 (and reply part for chapter III.)
    • Consider the integral of the Inexperienced’s operate for the pressured damped harmonic oscillator.
  • V0.1.17-2 (Feb 11, 2023)
    • Repair typos for the underdog swing equations (Inexperienced’s operate derivation lacking the damping fixed.) Due to prlw1 (on github.)
    • Change the elliptical parameterization in chapter II from the overly intelligent hyperbolic kind to simply weighted sine and cosines. The unique illustration given in an issue (resolution added.) Derivation of reciprocal body and verification added for the sinusoidal parameterization additionally introduced as an issue.
  • V0.1.16-19 (Feb 5, 2023)
    • Add options for many of the chapter 1 issues.
  • V0.1.16-16 (Oct 16, 2022)
    • Spelling and grammar error notes from Prof. Norman Derby
    • Make clear signal following eq. 1.23 — due to Prof Regular Derby!
  • V0.1.16-14 (Oct 9, 2022)
    • Present a touch for downside 2.1 (Wrenn Wooten.)
  • V0.1.16-13 (Could 15, 2022)
    • Repair equations 2.81 2.82 — error discovered by Christopher
    • check drove construct directions (slight repair required.)
    • restore latexsym Field as dAlembertian
    • gutter fixes.
    • perl -p -i -e ‘s/ *$//’ `cat spellcheckem.txt `
    • purge most dmath utilization. MacTex improve has made these seemingly malfunction, and features are getting cut up in all kinds of bizarre locations.
  • V0.1.16-5 (Sept 26, 2021)
    • N.D. Foreest: “in eqs 2.46 en 2.49 I’d present first learn how to compute the product of the 2 vectors. Then taking the scalar half provides the conclusion of two.46, however 2.49 turns into a lot shorter. Like this, the unifying thought of ga turns into additionally a bit clearer maybe”
  • V0.1.16-4 (Sept 22, 2021)
    • Peter Eriksen: I’ve discovered some small, potential errors within the guide:…
  • V0.1.16-3 (Aug 4, 2021)
    • Nicky, D, Foreest: typo in eq 2.134. Shouldn’t the operator L be in entrance of the second step?
    • Repair typo in vector Inexperienced’s operate for area time gradient
  • V0.1.16-1 (Feb 18, 2021)
    • Change a pair figures to black and white.
    • Parameterize the trail to twoParameterDifferentialFig.eps (colour vs. black and white.)
    • d’Lambertian -> d’Alembertian (Due to Jack Paladin!)
    • s/Othogonal/Orthogonal/ (due to Zhengbang Zhou)
    • Spelling fixes.
    • Run the duplicate phrase checker from
    • Ran hyperlink checker and stuck two.
  • V0.1.16-0 (July 28, 2020)
    • preface tweak: show Maxwell’s equation in closing kind from the get go.
    • Apply suggestion from sigs.
    • Repair errors in cross product wedge relation instance: due to Invoice Ignatiuk!
    • Rework formsVsGA.tex as an appendix.
    • David: typo: web page 18, (1.11) For instance… (e1)^2 = (e2)^2 = (e3)^3 = 1. The facility for (e sub 3) must be 2
    • Bruce Gould: pg 106: Within the final expression I believe the v1’s must be U2’s.
    • Add Tim Put and another reviewers that I’d missed to the thanks.
    • Try to repair the lacking dashes within the bash lstlisting.
    • Merge pull request #1 from TimPut/typos
  • V0.1.15-6 (Could 2, 2019)
    • Replace figures (thicker traces, take away some ticks, …) and hyperlink them to the mathematica hyperlink anchors.
    • “in determine fig.” -> “in fig”.
    • Prolong my hacks of the traditional thesis template to make use of 6×9 with smaller than default margins. Now have the preface web page numbers not within the bleed space of the web page.
    • Cut up colorlablebox into separate .sty (for phy452 notes.)
    • Repair pdfbookmarks for contents and checklist of figures (in order that they don’t present up below the preface)
    • Index quaternion (Bruce Gould)
    • GAelectrodynamics.tex: Need scrheadings beginning earlier than contents in any other case web page numbers are out of bounds (and the web page headings are MIA)
    • Bruce: “Could I counsel that the proofs ought to have the end-of-proof image on the finish?” Used the amsthm proof atmosphere to do that.
    • Theorem 1.2: flip the converse right into a footnote, to be seen later. (Bruce)
    • Added Bruce Gould to the thanks.
    • This model uploaded to kdp (successfully the “2nd” version)
  • V0.1.14 (Jan 2019)
    • varied edits to chapter 1, plus changes to supply a 6×9 model for createspace.
  • V0.1.8 (July 2018)
    • begin with correct definion of line integral
    • outline a bivector valued multivector floor space integral, and supply some examples.
    • quantity integral and instance
    • name the road integral end result: the elemental theorem for line integrals.
    • rework floor integral content material.
    • Bought the fundamental reorganization of chapter2 completed, correctly introducing line integral, floor integral and quantity integral earlier than the elemental theorem specializations.
    • rewrite normalVectors as theorem-example-proof
    • incorporate extra suggestions from Mo.
    • merchandise.tex: cut up the large hybrid definition into two theorems (dot is scalar choice, vector product has grades 0, 2) and one definition (wedge product).
    • fixes for instance: merchandise of two unit vectors.
    • use (for instance) “grade (0,1)-multivector” as a substitute of “0,1 multivector” or “grade 0,1 multivector”
    • Rework twin as definition, examples, dialogue, as a substitute of the definition final.
    • incorporate my annotated feedback as much as web page 12 (ch1).
  • V0.1.7: (April 2018)
    • Extra feedback from Mo. take away the determine that got here out clean within the printed model.
    • multivector definition modified to sums of scalars, vectors, and merchandise of vectors, particular circumstances of that are k-vectors. (How you can interpret merchandise equivalent to e_1 e_1 might want to come later.)
    • removed contraction axiom reference within the multivector/multivector area intro.
    • cut up out definition of multivector from multivector area. nonetheless enhancing the end result, which isn’t fully coherent.
    • rewrite preface once more.
    • intro supplies: add examples.
    • begin splitting the large ch1 abstract desk, and do a greater introduction to vectors to start out issues off.
    • add indexing of assorted symbols at level of first use.
    • change tables to make use of newtcolorbox (this may embed the specified format attributes).
    • small areas (,) after dV, dA, …. when they’re first within the integrand.
    • begin incorporating my paper edit notes from first draft createspace print.
    • experimenting with 6×9 format. didn’t cut back the createspace value (however elevated it)
    • lr’s on some vector derivatives.

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