From Maxwell's Equations and Hamilton's Quaternions to Vector Calculus

Battles over notation are some of the best (and most acrimonious) of battles!

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What I find so disturbing about Heaviside’s “murderous” dislike of potentials is that the electric and magnetic fields both derive from the magnetic vector potential “A”:

E=∂A/∂t (partial time derivative)
B=∇xA (curl)

If you have the wave equation for A, you automatically have both. In fact, this is how I learned electromagnetism and it is also how Carver Mead proposes physics should be taught so as to incorporate modern understanding of what he calls “collective electrodynamics”.

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A more concise and complete form for electromagnetism in terms of potentials appears as a subexpression of the Lorentz force:

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If you strip off the charge, you get an expression for a “motionalE or motional electric vector field which depends on a velocity vector field v. The expansion of the total derivative of A with respect to time (dA/dt) contains the subexpression vx∇xA, which is the magnetic portion of the Lorentz field.

It is really quite a puzzle to me that people don’t work with this definition since it contains all the components necessary to calculate everything you need for EM.

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I don’t know how many people around Scanalyzer follow Cormac McCarthy novels, but his most recent is inspired by the metaphysics of QM and contains this rather curious (to me) technical flaw:

Profound equations are often said to be beautiful Maxwell I suppose. If you overlook the E and B vector potential in place of the A.

This is echos, almost exactly, my prior comment except that the proper wording would have been “E and B fields in place of the A vector potential”.

This error confuses what may be a portal into the metaphysics he addresses since in my impression of the boundary between the paranormal and physics, it is the A vector potential that looms the largest.

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A History of Vector Analysis: The Evolution of the Idea of a Vectorial System”, by Michael J. Crowe, originally published in 1967. Dover edition published 1994, ISBN 0-486-67910-1, 270 pages.

Kathy, who loves physics, strongly recommended this book. Ah! For those happy days half a century ago when authors could write and editors could edit. Kathy is right – this is a well-written interesting book, definitely worth reading.

Vectors are mathematical entities with both magnitude & direction. Vectorial analysis can be traced back to the late 1700s when Gauss began to play around with representing complex numbers on a plane. The pace picked up around 1843 when Sir William Hamilton invented quaternions. Eh? What on Earth is a quaternion? For that, we have to jump forwards about half a century to the view of that acerbic self-taught genius Oliver Heaviside: “A quaternion is neither a scalar, nor a vector, but sort of combination of both. It … is a highly abstract mathematical concept”.

Hamilton himself was a certifiable genius who was appointed professor at Trinity College, Dublin while still an undergraduate. His justifiable fame resulted in his quaternions getting a reasonable amount of notice in mathematical circles. At almost the same time as Hamilton, Herr Grassmann, a humble German schoolmaster, independently created a vectorial system. However, academic outsider Grassmann was almost totally ignored.

Things really began to heat up in 1873 when the great James Clerk Maxwell published his theory of electricity & magnetism which was based on the use of “fields” to explain action at a distance. Now, action at a distance is one of those concepts which becomes discomforting if one thinks about it too closely – but Maxwell’s theory was based on fields, which required him to make some use of the mathematics of Hamilton’s quaternions.

Nearly two decades later Heaviside entered the fray and poo-pooed quaternions. He stripped Maxwell’s equations down to a modern vectorial form. Meanwhile, Willard Gibbs at Harvard – who is more commonly remembered these days for his work on thermodynamics – independently and simultaneously also generated the modern form of vectorial analysis.

Over the period 1890 – 1894, a mathematical bun fight broke out between the supporters of quaternions and those who preferred the emerging Heaviside-Gibbs formulation. Much ink was spilled on such topics as the peculiarity that the square of a quaternion had to be a negative number.

By 1910, physicists had concluded that the Heaviside-Gibbs vector formulation was more useful for dealing with real physical issues such as optics and electricity & magnetism. Academic debates over whether vectors should be represented by Gothic letters or Greek letters were resolved, and the modern system of vectorial analysis emerged. Herr Grassmann’s prescient contributions were still largely ignored. As the author notes: “Grassmann was unknown however and suffered the fate of a man whose first great discovery is revolutionary”.

The reader will not learn much about vectors from this book, but will enjoy a lot of fascinating history about the development of mathematics and physics.

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When quaternions were originally developed, many mathematicians thought them extremely strange because, unlike real and complex numbers, multiplication and division of quaternions is not commutative. If \bf a and \bf b are arbitrary quaternions, except for a few special cases in which they reduce to less general number systems:

{\bf a} \times {\bf b} \neq {\bf b} \times {\bf a}

It was realised that quaternions were equivalent to rotation in three-dimensional space, with multiplication equivalent to first performing one rotation, then another. These operations do not commute either, and in fact behave precisely as quaternions do. (Think about it: if you have a cube and rotate it first around its X axis by 90° and then around its Y axis by 90 degrees, it will end up in a different orientation than if you first rotate in Y and then X. (Until you become comfortable with 3D rotations, this stuff may make your head explode. The Fourmilab Orientation Cube can help you grasp what’s going on: here is a video demonstration.)

With the development of vector analysis, quaternions were largely displaced in physics and engineering by vectors, and were considered an arcane topic in abstract mathematics. All of this changed in the latter half of the 20th century with the advent of three-dimensional computer graphics, robotics, and spacecraft control and navigation, where quaternions allowed one to easily do mathematics on 3D rotations without the Hellish mess of trigonometric functions and risk of degeneracies and singularities that lurk when doing it the “easy way”.

Today, most extensible application programming languages such as Python, Rust, Perl, Mathematica, and even C++ have packages that implement quaternion arithmetic.

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I first learned to use quarternions as practical tools more than twenty years ago, programming ABB industrial 6-axis robots.

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