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I think the essay is not about Maxwell's theory being hard for college students to understand, but rather, for other 19th century physicists to understand. And that's mainly because Maxwell himself didn't do a very good job of communicating his theory at the time, so it took other talented physicists to rework, explain, and popularize his ideas.

People forget that Maxwell's Theory contradicts the prevailing belief of the time that waves ALWAYS require a medium for propagation--the so called "aether".

That's a really large conceptual jump and physicists did not make that jump lightly or easily.

Maxwell's paper was 1865. The Michelson–Morley experiment was 1887. Michelson himself couldn't make the shift. Quoting Wikipedia (https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_exper...): "The negative result led Michelson to the conclusion that there is no measurable aether drift.[1] However, he never accepted this on a personal level, and the negative result haunted him for the rest of his life (Source; The Mechanical Universe, episode 41[8])."

In an attempt to preserve "aether", Lorentz contraction then enters the picture as an ad hoc explanation for the Michelson-Morley result. It turns out Lorentz contraction is correct, but not because of the existence of "aether" but because of the constancy of the speed of light--c (Einstein Special Relativity--1905).

Once you finally give up on "aether" after 40 years of trying otherwise, you can finally just roll with the mathematical implications of Maxwell's equations.

Maxwell saying that his theory "attributes electric action to tensions and pressures in an all-pervading medium... the medium being identical with that in which light is supposed to be propagated" suggests that Maxwell himself did not view his theory to be contradicting the existence of an aether (or was being coy about it). Which is especially interesting because Maxwell based his theory on Faraday's, and Faraday didn't believe in the aether.[1]

[1] Michael Faraday's Thoughts on Ray Vibrations, 1846, cited by Maxwell in his paper. Faraday says: "The view which I am so bold to put forth considers, therefore, radiation as a kind of species of vibration in the lines of force which are known to connect particles and also masses of matter together. It endeavors to dismiss the aether, but not the vibration." (https://pwg.gsfc.nasa.gov/Education/wfarad1846.html)

I mean, language is hard, and it evolves. What they called mediums, we now call (quantized) fields. QED would say the photon field _exists_ at all points in space and that it is kinda fair to say that it is a "medium" for electromagnetic forces and waves to propagate (with the additional point that special relativity is required to understand how to correctly transform it into a different reference frame, rather than imagining it as a strictly Newtonian medium).

Admittedly the tools to make the mathematics more compact and clearer weren't invented yet (except as quaternions).

We did get stuck with Gibbs' vector calculus formulation as the canonical view unfortunately.

Which is interesting considering Freeman Dyson was the guy that made the connection between Schwinger's and Feynman's QED.

Are there any textbooks you would recommend for learning vector analysis / vector fields before studying EM?

EM’s vector fields formulation is fairly straightforward: it’s all curls and divergence. Any 3rd (-ish) semester undergrad multivariate calculus course is likely to cover it in sufficient depth. “Mathematical Methods for Physicists” covers it in sufficient depth, for example, provided you already have a thorough understanding of the prerequisite material. Most undergrad physics degree curriculums should have E&M courses whose texts (e.g. Introduction to Electricity and Magnetism, Griffiths) cover enough of the details. If you want to pursue it further than an undergraduate level study, you’ll also want a good text on differential equations that has or is supplemented by material covering, e.g., spherical harmonics and Bessel functions (among other things). I wish I could remember what I used, but it was…more years ago than I care to say when I was an grad student.

I do not. None of the books I know of are very good because they are mostly targeted at Mathematics majors rather than physicists or engineers.

Gerard ’t Hooft used to have a humongous list of textbooks for aspiring theoretical physicists. I'd sure look there for starters.

I was recommended Nathan Ida's Engineering Electromagnetics as being comprehensive in that all the necessary Mathematics is introduced in place as needed. Lookup the reviews for this book on the web.

Perhaps somebody who has read this book can comment in more detail.

I've always thought the Heaviside notation is a bit bizarre... is there any advantage to them at all?

Heaviside and his proponents avoided quaternion like a plague, and like a classic false messiah he somehow convinced people not to use it. If we want to easily and completely model the EM waves its entirety including polarization we need to embrace quaternion, there is no two ways about it. The intuitive understanding of EM can be only developed by using quaternion and me personally waiting for someone to write a Pozar's book version in quaternion approach.

Vector fields are just as incorrect, and ought to be relegated to history books as a mere stepping stone on the way to understanding.

Geometric algebra can reduce Maxwell’s equations into a single, hilariously terse equation:

    ∇F = J

Ref: https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...

Thanks for the pointer on Geometric Algebra. This looks to be a promising path to understanding relativity/QM/EM, and goes some way to explaining my unease with cross products and imaginary numbers.

Disclaimer: maths degree, so my unease was not a plain lack of understanding.

> maths degree

Then you want differential forms for EM, differential geometry more broadly for GR, and a bit of functional analysis for QM.

The hype around geometric algebras (Clifford algebras over R) just comes from the fact that it's not the plug'n'chug explicit numbers and coordinates approach, which is all most people ever see. They do not do a good job of tracking the physical structure of electromagnetism, and in fact end up baking in a lot of assumptions about the setting that fail to generalize.

> baking in a lot of assumptions about the setting that fail to generalize

That’s the point! That’s the entire point!

Mathematicians want the most general, most abstract approach. They want to generalise to a wide range of problems and not be painted into any one specific example.

Physics theories have an opposite goal to this: the ideal theory ought to take no parameters, and produce “reality” as the one and only possible outcome. The ideal theory ought not generalise to un-physical models.

For example, the mathematics of general relativity have excess degrees of freedom that must be constrained through additional restrictions. Similar issues turn up almost anywhere matrices are used: they have too many degrees of freedom.

Geometric Algebra is typically a better fit for what actually goes on in physics.

For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated. Rotations implemented using GA have none of these issues.

> The ideal theory ought not generalise to un-physical models.

Sure - but a theory that fails to generalize to physical models is a bad one. A good classical theory should be a straightforward deformation of the corresponding quantum and/or relativistic theory. In this respect the best versions of classical mechanics are the standard Lagrangian and Hamiltonian approaches.

> For example, rotation matrices have precision issues, gimbal lock, and can’t be robustly interpolated.

No physicist was ever under the impression that euler angles were any realer than any other way of parameterizing SO(3), that matrices were the linear transformations they represent, that manifolds are their charts, or any other trivial map-territory confusion. Paying careful attention to the distinction between real physical objects on the one hand and their representations on the other is the central theme of the last century of physics: that's basically all a gauge theory is!

This is exactly what I'm talking about with geometric algebra advocates only ever comparing it to the worst sort of high school coordinate-bashing imaginable. The argument always goes

- Look at this horrible vector algebra with explicit charts and numbers all over the place

- Now look at this nice coordinate-free geometric algebra construction

- Therefore geometric algebra is the right setting for physics

but it's a complete non sequitur, because the coordinate-freeness is doing all the heavy lifting. But everyone already works without coordinates wherever it's practical to do so! The question isn't whether you should work in terms of abstract objects or explicit coordinates, it's which abstract objects you should work with.

Thanks for your comment.

> They do not do a good job of tracking the physical structure of electromagnetism

What do you mean by tracking the physical structure?

By this, do you mean the typical EM formulation of Maxwell's laws produces Gauss's law and Faraday's law (which are instructive) where as the Geometric Algebra formula produces ∇F = J (less instructive?).

> and in fact end up baking in a lot of assumptions about the setting that fail to generalize.

Can you explain a bit more what you mean here please?

> By this, do you mean the typical EM formulation of Maxwell's laws produces Gauss's law and Faraday's law (which are instructive) where as the Geometric Algebra formula produces ∇F = J (less instructive?).

> Can you explain a bit more what you mean here please?

It's all downstream of geometric algebra leaving the metric implicit in its operations, basically

- The metric is an extremely important physical quantity: it doesn't necessarily look that way when everything is classical and flat, but you have to start caring about it in curved spacetimes.

- Even when the metric can be safely neglected, doing so makes it very easy to confuse degree (n-k) elements of your vector space V with degree k elements of its dual V. V and V are isomorphic but not canonically isomorphic - you have to choose a basis. And keeping track of where you introduce a choice of basis matters, because all physical quantities are basis-independent. Nature has no preferred coordinate system.

- Geometric algebra doesn't make sense on a general manifold: you have to embed it in a sufficiently large geometric algebra and inherit structure from the embedding. This is better than working in a particular coordinate chart but worse than doing differential geometry in a purely geometric way.

- Geometric algebra is not invariant under diffeomorphism, which means GR is dead in the water. You can build mostly equivalent theories by enforcing the equivalence principle in your dynamics instead, but it's more complicated and ironically far less geometric.