ChapterZero

Did you know that the 4 equations known as Maxwell’s equations are actually Heaviside’s equations? Maxwell’s equations were actually 20 equations in 20 variables. This is my fourth year as an EE student, and I have never heard this before; this is like having the rug pulled from under my feet— I emailed one of the professors in the EM group to see if he can clarify this for me. Shocking!

I stumbled across that fact while looking for information on Heaviside’s operational calculus, part of my intellectual heritage as an electrical engineer This was a method Heaviside developed during his stint as a telegrapher to solve ordinary and partial differential equations, based upon regarding differentiation as an operator. It has subsequently been displaced by the Laplace transform, even though his unorthodox approach (i.e. regarding division by the differentiation operator as integration!) has been given a solid basis by Bromwich. Here’s an example, taken from “Fourier Analysis and Boundary Value Problems” by Gonzalez-Velasco:

… to solve the simple differential equation for subject to initial conditions , Heaviside would denote the differential operator by — he had used the letter until 1886— and obtain the equation , where denotes the function that vanishes from and has value for [The infamous Heaviside step function]. Hence, if we treat as an algebraic quantity,

.

This cannot be the end of the line, of course. To obtain the actual solution from this operational solution, let us accept that the geometric series expansion is valid for this operator, and then

.

Since represents differentiation, Heaviside regarded as integration from zero to . In this manner,

and

for any positive integer , leading to the actual solution

,

which, perhaps surprisingly, is the correct solution.

I’ll say that’s surprising! Here’s a germane quote: shall I refuse my dinner because I do not fully understand the process of digestion? Well, I guess not.

On a related note, “Fourier Analysis and Boundary Value Problems” is a surprisingly good book— it is tailored more for engineers than mathematicians, I would say, yet it contains rigorous Hilbert space concepts, proofs, and even an introduction to measure theory (in the problems of Chapter 2), although Reimannian integration is used throughout, with a series of proofs given in an appendix to plug all the holes that were sprung from deserting Lebesge world. It doesn’t hurt that the typography is very well done, and there are lots of interesting, as opposed to forced, historical interludes. Too bad I’m borrowing it from a friend, instead of owning it.