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I read somewhere, some time ago that Maxwell originally wrote his eponymous equations using the formalism of quaternions and it was only the later intervention of Gibbs and Heaviside that put them into the modern form, that is via vector analysis (I say modern, but its more modern to write them via differential forms or Clifford analysis).

Is this true? And if true, how many equations were there in that formalism?

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3 Answers 3

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I read somewhere, some time ago that Maxwell originally wrote his eponymous equations using the formalism of quaternions ... Is this true?

It seems that the answer is "Not quite". Maxwell originally wrote his equations in components, and later simplified them by using quaternions and some vector calculus. It is true that Heaviside and Gibbs put them into their modern form.

And if true, how many equations were there in that formalism?

My source says 20. I count 12, but it says that "cont. eq. missing here", so maybe there were 8 equations relating to continuity.

Source: slides from the presentation given at University College London in 2010 On the changing form of Maxwell’s equations during the last 150 years β€” spotlights on the history of classical electrodynamics by Friedrich W. Hehl.

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  • $\begingroup$ There are 20, related $\endgroup$
    – Mauricio
    Aug 5 at 13:31
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Maxwell's equations as they are expressed today, with modern vector notation and not Maxwell's quaternion notation, were first written by Oliver Heaviside in a few papers in the mid- to late 1880s, but it wasn't until 1893 that the first volume of his Electromagnetic Theory appeared (and vol. 2 in 1899 and vol. 3 in 1912).

See:

Also, see this modernization of Maxwell's version of the Lorentz force law.

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The 1861 paper primarily used differential forms, but in the context of line integrals and area integrals rather than as a Grassmann calculus, though it was around at the time (Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik; i.e. The Theory of Linear Extension, a New Branch of Mathematics, 1844). I qualify that, since there was at least one place, later, in the treatise where the anti-commuting rule $dx dy = -dy dx$ popped up.

The 1864 paper used differential forms less so; the 1873 treatise used a pre-Heavisde form of vector calculus cannnibalized from Hamilton's quaternions ... only sparingly, to present the equations in capsule summary form.

You really didn't need to ask, since the treatise is on-line.

Volume 1: https://archive.org/details/electricandmagne01maxwrich
Volume 2: https://archive.org/details/electricandmagne02maxwrich

The equations are in Chapters VIII and IX of volume 2.

Here are the quantities and equations posted in the 1864 paper. I will reconcile them against what would best pass for the contemporary 3-vector form of the same today.

The Quantities:

  • Electromagnetic Momentum: $(F,G,H)$, $F dx + G dy + H dz$. Today: $𝐀 = (F,G,H)$.
  • Magnetic Intensity: $(Ξ±,Ξ²,Ξ³)$, $Ξ± dx + Ξ² dy + Ξ³ dz$. Today: $𝐇 = (Ξ±,Ξ²,Ξ³)/(4Ο€)$. He's got that extra $4Ο€$ in all his writings and it clouds his equations and analyses.
  • Electromotive Force: $(P,Q,R)$, $P dx + Q dy + R dz$. Today: $𝐄 = (P,Q,R)$, but only at 0 velocity. Our $𝐄$ is his 0-velocity $𝐄$, while his generic $𝐄$ (which we could designate $𝐄_M$) would be our $𝐄_M = 𝐄 + 𝐆×𝐁$, where $𝐆$ denotes the velocity. A distinction is made in the 19th century literature between the "stationary" and "moving" forms of Maxwell's equations - more on that below.
  • Current Due To True Conduction: $(p,q,r)$, $p dy dz + q dz dx + r dx dy$. Today: $𝐉 = (p,q,r)$.
  • Electric Displacement: $(f,g,h)$, $f dy dz + g dz dx + h dx dy$. Today: $𝐃 = (f,g,h)$.
  • Total Current: $(p',q',r')$, $p' dy dz + q' dz dx + r' dx dy$. Today: $𝐂 = (p',q',r')$. This notation is mostly defunct today, though it does pop up in a few places (e.g. A.O. Barut's "Electrodynamics and Classical Theory of Fields and Particles"). This is the current that is used in circuit analysis, not $𝐉$.
  • Quantity Of Free Electricity: (per unit volume): $e$, $e dx dy dz$. Today: Our $ρ$ = his $e$.
  • Electric Potential: $Ξ¨$. A scalar (so as a differential form, it is still just $Ξ¨$). Today: Our $Ο†$ = his $Ξ¨$.
  • Magnetic Induction: $(ΞΌΞ±,ΞΌΞ²,ΞΌΞ³)$, $ΞΌΞ± dy dz + ΞΌΞ² dz dx + ΞΌΞ³ dx dy$. Today: $𝐁 = (ΞΌΞ±,ΞΌΞ²,ΞΌΞ³)$. He does not make a clean distinction between Magnetic Induction and Magnetic Intensity in 1861 or 1864, but still makes an effective distinction, writing the terms as diglyphs as I did here, when the sense Magnetic Induction is intended. This was a huge mistake! As you can see, the two are completely different types of objects. Because of this confusion, he fails to note that $𝐁$ is a 2-form. The gap is only partly rectified in the treatise - where a distinction is made. But the failure to distinguish between the two, still leads to an erroneous equation (even in the treatise), which Thomson later had to correct. More on that below.
  • Coefficient Of Induction: $ΞΌ$. Today: our $ΞΌ$ is his $4πμ$. The extra $4Ο€$ was moved from his $𝐇$ to our $ΞΌ$. That change post-dates the treatise. I don't know who did it.
  • (Unnamed electric coefficient): $k$. Today: $Ξ΅ = 1/k$. He made the change by the time of the treatise. He said it's analogous to a "spring coefficient", while $ΞΌ$ is analogous to mass, in the spring force law.
  • Conductivity: $ρ$. Today: our $Οƒ$ is his $-ρ$. I don't know if the sign flip was intentional or an error, because it's not the only sign error he made.
  • The Velocity: $(dx/dt, dy/dt, dz/dt)$. Today: $𝐆 = (dx/dt, dy/dt, dz/dt)$, mostly defunct. In the 1873 treatise, he used $𝐆$ or $𝐯$ - sometimes even in the same section. The distinction between the "stationary" and "moving" forms of Maxwell's equations hangs entirely on whether $𝐆 = 𝟎$ or $𝐆 β‰  𝟎$. It's just left dangling there in the equations and he doesn't say what it's with respect to. But the context makes it clear: the stationary form applies to the unique frame of reference in which isotropic constitutive laws exhibit their isotropy - the unique frame in which $𝐃$ & $𝐄$, and $𝐁$ & $𝐇$ are each proportional. In this frame, an electromagnetic disturbance propagates outward with a fixed center. In all other frames, $𝐆$ indicates how fast the center would move. This frame is unique in both non-relativistic theory and in Relativity in the presence of a medium. It is not unique in Relativity in a vacuum, and in that setting $𝐆$ is arbitrary and (mostly) superfluous. That was the main point of Einstein's 1905 "On the electrodynamics of moving bodies". The term "moving" was referring to the $𝐆 β‰  𝟎$ condition. His thesis was that (for the vacuum), one could take the stationary form $𝐆 = 𝟎$ - for all frames (and that the in-vacuum light speed is an absolute speed, not relative at all)! Einstein was motivated partly by the question: what if $|𝐆|$ is equal to the electromagnetic wave speed. Ironically, his 1905 paper didn't actually answer that question and (in fact) in the 1908 treatment, by him and Laub, where he returns to the issue of moving media, where a velocity, like $𝐆$, returns once again, the question comes back. They failed to notice that. It's still a dangling issue - all the way up to the present time! That's the issue of light-speed media.

The Equations:

  • (A) Total Currents: $$\left(\begin{align} p' = p + \frac{df}{dt}\\ q' = q + \frac{dg}{dt}\\ r' = r + \frac{dh}{dt} \end{align}\right) β‡’ 𝐂 = 𝐉 + \frac{βˆ‚πƒ}{βˆ‚t}$$
  • (B) Magnetic Force: $$\left(\begin{align} ΞΌΞ± = \frac{dH}{dy} - \frac{dG}{dz}\\ ΞΌΞ² = \frac{dF}{dz} - \frac{dH}{dx}\\ ΞΌΞ³ = \frac{dG}{dx} - \frac{dF}{dy} \end{align}\right) β‡’ 𝐁 = βˆ‡Γ—π€$$
  • (C) Electric Currents: $$\left(\begin{align} \frac{dΞ³}{dy} - \frac{dΞ²}{dz} = 4Ο€p'\\ \frac{dΞ±}{dz} - \frac{dΞ³}{dx} = 4Ο€q'\\ \frac{dΞ²}{dx} - \frac{dΞ±}{dy} = 4Ο€r' \end{align}\right) β‡’ βˆ‡Γ—π‡ = 𝐂$$
  • (D) Electromotive Force: $$\left(\begin{align} P = ΞΌΞ³ \frac{dy}{dt} - ΞΌΞ² \frac{dz}{dt} - \frac{dF}{dt} - \frac{dΞ¨}{dx}\\ Q = ΞΌΞ± \frac{dz}{dt} - ΞΌΞ³ \frac{dx}{dt} - \frac{dG}{dt} - \frac{dΞ¨}{dy}\\ R = ΞΌΞ² \frac{dx}{dt} - ΞΌΞ± \frac{dy}{dt} - \frac{dH}{dt} - \frac{dΞ¨}{dz} \end{align}\right) β‡’ 𝐄_M = 𝐆×𝐁 - βˆ‡Ο† - \frac{βˆ‚π€}{βˆ‚t}$$
  • (E) Electric Elasticity: $$\left(\begin{align} P = kf\\ Q = kg\\ R = kh \end{align}\right) β‡’ 𝐃 = Ξ΅ 𝐄_M$$
  • (F) Electric Resistance (Possible sign error): $$\left(\begin{align} P = -ρp\\ Q = -ρq\\ R = -ρr \end{align}\right) β‡’ 𝐉 = Οƒ 𝐄_M$$
  • (G) Free Electricity (Sign error): $$e + \frac{df}{dx} + \frac{dg}{dx} + \frac{dh}{dx} = 0 β‡’ ρ + βˆ‡Β·πƒ = 0$$ It should be ρ = βˆ‡Β·πƒ.
  • (H) Continuity: $$\frac{de}{dt} + \frac{dp}{dx} + \frac{dq}{dy} + \frac{dr}{dz} = 0 β‡’ \frac{βˆ‚Ο}{βˆ‚t} + βˆ‡Β·π‰ = 0.$$
  • (I, for implied) Magnetic Induction: $$\left(\begin{align} ΞΌΞ± = ΞΌΒ·Ξ±\\ ΞΌΞ² = ΞΌΒ·Ξ²\\ ΞΌΞ³ = ΞΌΒ·Ξ³ \end{align}\right) β‡’ 𝐁 = μ𝐇$$

Maxwell did not make clear distinctions between total and partial derivatives - because he was young and inexperienced; only in his 20's and 30's, dying in his 40's. His failure to make a clean separation between $𝐁$ and $𝐇$ led to his writing down the wrong (implied) equation linking the two. With the correction later added by Thomson, it would read: $$𝐁 = ΞΌ(𝐇 - 𝐆×𝐃).$$ Only in this way do you get consistency with the Galilean transforms; and his equations were meant to be covariant under Galilean transforms.

The $𝐂$ vector is superfluous. Equations (A) and (C) can be combined into (AC): $$βˆ‡Γ—π‡ = 𝐂 = 𝐉 + \frac{βˆ‚πƒ}{βˆ‚t} β‡’ βˆ‡Γ—π‡ = 𝐉 + \frac{βˆ‚πƒ}{βˆ‚t}$$

The alphabet soup for the contemporary names is Maxwell's doing and came from his treatise, where he used vectors for capsule summaries of the equations, much as I just did, above.

The sign error in (G) leads to a cascade of other oversights or errors. First, when it written correctly (as I did above), then it becomes clear that (G), along with (AC) together already yield (H); so that (H) is superfluous. Otherwise, the sign error would make (A), (C), (G) and (H) mutually inconsistent. The failure to note that (H) was superfluous ... and that there is one more field quantity (20) than equations (19) (or 23 versus 22, if you distinguish $𝐁$ from $𝐇$) ... leads to a failure to note that there is a partial redundancy in the quantities, by this account. That's the gauge symmetry, i.e. the invariance of the equations with respect to the following gauge transform: $$Ο† β†’ Ο† + \frac{βˆ‚Ο‡}{βˆ‚t}, 𝐀 β†’ 𝐀 - βˆ‡Ο‡.$$

In fact, Maxwell already partially acknowledged this by pointing out that $Ο†$ was subject to arbitrary constant adjustment. In his 1861 paper and in the treatise, he adopted the "Coulomb gauge": $βˆ‡Β·π€ = 0$; but he could have just as well used the gauge transforms to impose the gauge condition $Ο† = 0$, since he already said it was arbitrary and didn't really use $Ο†$ anywhere else.

The Coulomb condition is invariant under Galilei transforms, though $Ο† = 0$ is not. So, with a Galilei transform, you'd have to regauge $(Ο†,𝐀)$ to get back $Ο† = 0$ in the new frame of reference.

The expression for the field $𝐄_M$ is frame-dependent, though $𝐄_M$, itself, is not. It is best decomposed into our $𝐄$, which is frame-dependent. But this removes the dangling vector from the field equation. The pair $(𝐁,𝐄)$ fit more consistently together than do $(𝐁,𝐄_M)$.

There are no equations for $βˆ‡Β·π = 0$ or $βˆ‡Γ—π„ + βˆ‚π/βˆ‚t = 𝟎$, because the first one is already implied by (B), and the prospect for the second one is obscured by the fact that Maxwell was using $𝐄_M$, instead of $𝐄$. Both appeared in the 1861 paper, the second being asserted only for the stationary case $𝐆 = 𝟎$, where $𝐄_M$ reduces to $𝐄$.

The failure to make this separation also obscures the frame-dependency in (F) and its inconsistency. The vector $𝐉$ is frame-dependent, while $𝐄_M$ is not! (OOPS!) So, actually (F) also selects out a frame of its own, "The Frame Of Resistivity" if you will. Moreover, it need not be the same as the "Frame Of Isotropy". That's also obscured by the use of $𝐄_M$ in (F).

Einstein and Laub correctly noted (and fixed) the frame dependency of (F), as did Minkowski in all their 1908 treatments of the theory of moving media. But they both failed to note that the frame for (F) doesn't have to be the same as the Frame Of Isotropy. So, it may use a different $𝐆$ vector, $𝐆'$. If it is rewritten to fix the inconsistency, it would have the form: $$𝐉 - 𝐆' ρ = Οƒ (𝐄 + 𝐆'×𝐁).$$

So, after making Thomson's correction to the unnamed extra equation, removing the sign error from (G), eliminating the now-redundant (H), removing the largely-superfluous $𝐂$ and splitting $𝐄_M$ into $𝐄$ and $𝐆×𝐁$, and repairing (F), the result, with a few extra adjustments by moving around terms, is the following system:

  • (B): $𝐁 = βˆ‡Γ—π€$
  • (D'): $𝐄 = -βˆ‡Ο† - \frac{βˆ‚π€}{βˆ‚t}$
  • (AC): $βˆ‡Γ—π‡ - \frac{βˆ‚πƒ}{βˆ‚t} = 𝐉$
  • (G'): $βˆ‡Β·πƒ = ρ$
  • (E): $𝐃 = Ξ΅ (𝐄 + 𝐆×𝐁)$
  • (I'): $𝐁 = ΞΌ (𝐇 - 𝐆×𝐃)$
  • (F'): $𝐉 - 𝐆' ρ = Οƒ (𝐄 + 𝐆'×𝐁)$

The equations (B), (D'), (AC), (G') are diffeomorphism covariant and can be expressed entirely in terms of 4D differential forms. They are "paradigm neutral", since geometry (at the level of differential manifolds) can't tell the difference between relativistic theory and non-relativistic theory, nor (for that matter) even between space-like and time-like dimensions. The equations apply equally well - as is - to both relativity and non-relativistic theory.

The equations (E) and (I') are the constitutive laws and are where all of the difference between paradigms, and all the dependency on the space-time metric are encapsulated within. They are not only different between relativity and non-relativistic theory, but also between flat space-time and curved space-times. They may be referred to as the Maxwell-Thomson Relations, while their relativistic forms, which are respectively equivalent to the following: $$𝐃 + \frac{1}{c^2} 𝐆×𝐇 = Ξ΅ (𝐄 + 𝐆×𝐁)$$ $$𝐁 - \frac{1}{c^2} 𝐆×𝐄 = ΞΌ (𝐇 - 𝐆×𝐃)$$ are what Einstein and Laub derived in 1908, as did Minkowski in 1908 in the paper where he introduced his 4D chrono-geometry. They are known as the Maxwell-Minkowski Relations.

If $Ρμ = 1/c^2$ (i.e. a vacuum), then the Maxwell-Minkowski relations are equivalent to the "stationary form" $𝐆 = 𝟎$, regardless of what $𝐆$ is ... provided that $|𝐆|$ is less than the wave speed $V = 1/\sqrt{Ρμ}$ and $|𝐆| < c$. If not, then things get complicated ... and nobody's ever addressed that issue even up to the present day, to the best of my knowledge.

So, $𝐆$ is "superfluous", to use Einstein's 1905 characterization, but not entirely. The irony is that the "not entirely" part pertains to the very matter that motivated his research: the case where "you're moving along with the light wave", e.g. where you're in a medium whose frame of isotropy is moving at the wave speed of the medium. You might call that the "Cherenkov Threshold". Moreover, the equations restricted to that setting even have a non-trivial vacuum limit. In the limit, there is still a vestige of the $𝐆$ vector. Nobody's ever noticed that, as far as I'm aware.

Finally, the equation (F') also has a relativistic form, but Einstein & Laub and Minkowski all wrote it with the tacit assumption that $𝐆' = 𝐆$: i.e. that The Frame Of Resistivity = The Frame Of Isotropy. That was an oversight. With the vectors kept distinct, the relativistic version would be: $$𝐉 - 𝐆' ρ = Οƒ \frac{𝐄 + 𝐆'×𝐁 - 𝐆'𝐆'·𝐄/c^2}{\sqrt{1-|𝐆'|^2/c^2}}.$$

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  • $\begingroup$ TL;DR. Etc etc etc. $\endgroup$ Aug 11 at 7:44
  • $\begingroup$ That's the way we like it! So actually: NLE;RIA. $\endgroup$
    – NinjaDarth
    Oct 22 at 18:52

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