Maxwell did not use vector calculus, which actually makes it pisser to understand because it bypasses having to convert the vectors to coordinates. I find short hand notion more confusing than writing it all out even if the latter takes more room. I did not understand general relativity at all until I saw an example where everything was written out and then it made much more sense
This is something of a myth. Maxwell actually uses both to ease the mathematical pain for his readers - he writes the equations out coordinate by coordinate, and he also compresses them using Hamilton's (quaternion) model of vectors. In fact Maxwell was the one who introduced the gradient, curl, and divergence (actually he used convergence) operators, and he addressed precisely your concern in his book by giving both formulations!
Maxwell used per coordinate equations and quaternion equations, but never wrote out what are now known as "Maxwell's equations," which were first formulated by heaviside using Gibbs-Heaviside vector calculus.
Gibbs-Heaviside vector calculus formulations proved far more valuable to engineers and scientists than the equivalent quaternion formulations - despite the criticisms from Hamilton and Tait.
Maxwell explicitly uses the imaginary part of a quaternion as a model of vectors, just as Hamilton did:
> A german letter denotes a Hamilton vector, and indicates not only its magnitude but its direction. The constituents of a vector are denoted with roman or Greek letters. [1]
Gibbs and Heaviside or course use a different algebraic model of vectors, which is the one used in introductory courses now. But the more significant change made to get to "maxwells equations" as we know them now is not a slight change in the mathematical formalism.
The big difference is a change in the underlying physical model: Maxwell's original treatment is largely in terms of potentials, which Heaviside modified to focus on the E-M force fields directly. This change is what gives the familiar four equations we know today, not fiddling with your parameterization of the rotation group, or similar games. Nowadays physicists frequently use a tensor or forms model of vectors in addition to the gibbs model.
I dont think I agree. Yes they are not the Gibbsian vector but they are a pretty damn good model of what we want to model as vectors. Its already a little closer to geometric algebra / calculus than Gibbsian vectors. Its a pity that Grassman and Clifford's work gained attention much later.
Once javascript is out of the bag one cant do much about it. Thats how I feel about Gibbsian vector calculus and geometric algebra. Not that javascript or Gibbsian vectors are bad, quite the contrary, they are astonishing in their own right and scope, but one still feels they could have been so much better.
I think geometric algebra is a very cool and elegant approach as well, but it has not yet proved useful or valuable to working engineers.
If we want to compare to programming languages, perhaps Haskell vs. C. One is beautiful and elegant, the other is down to earth and conceptually simple for "normal" people. One is primarily used by academics and researchers, the other is used to build the world. Both are admirable.
They would have, had they been the "first mover". Hence my Javascript analogy
Had a better language been released at that time it would have been just as useful and valuable. In fact much more so. But as is usual, a "worse_is_better" first mover accrues so much head start in the mind share and in the tooling / literature that it becomes impractical to switch, especially when both can express the same set of things (just that one does it more concisely than the other). Another shallower example: imperial vs metric units.
I dont think its like Haskel::C at all. Haskell is way more mathy and has a steeper learning curve than C. One who can master curl divergence and vector stoke's theorem can master geometric algebra with less effort
> not yet proved useful or valuable to working engineers
The people who value it are those who must deal concretely with the details of the model on a daily basis: computer programmers working in graphics, computer vision, robotics, physical simulation, ...
It doesn’t do all that much for many pure mathematicians who are happy to hand-wave away the fine details content with the knowledge that the languages can be proven equivalent by someone else, just like it doesn’t do much for the type of engineers who just use someone else’s software for their work.
Of course, the current mess is a major point of pain/confusion for the generations of undergraduates studying vector calculus. But they should go through the same hazing ritual the rest of us did, right? It’s only fair.
