I am the advisor to a high-school amateur radio club. The students in this club are curious about quite a number of mathematical and physical questions that neither they nor I have been able to get satisfactory answers to.
A typical thing that I find is that the explanations are oriented way too much toward experts (Wikipedia is a prime offender here), or somewhat less too much for experts, but still too much (Weisstein et. al.), or are incomprehensible explanations (many web sites), or way too fragmented and unhelpful (everything else.)
Textbooks don't go into this material at the level that these people need (high-school and some college level books), or are way to advanced (other college level books.)
A lot of the problem may be related to the arcaneness of the subject matter in amateur radio and electronics. Ideally, what these people need are reference materials written by people who are both knowledgeable in the theory, and familiar with its applications to real-world problems.
Some (but by no means all) of the questions relate to linear algebra, and, seeing that there is some interest in explaining linear algebra here at Conservapedia, I thought I would ask.
After brainstorming with the students about what they would like to see, we came up with these.
What, intuitively, do the divergence and curl operations mean? How do you show that they are coordinate system invariant? In fact, is there a good way to show that even the cross product is invariant? There ought to be some good physical intuition here.
What, intuitively, do Maxwell's equations say about divergences and curls? How should we visualize electromagnetic fields and waves in terms of those divergences and curls?
How is it that electromagnetic waves are a solution to Maxwell's equations?
How is it that transmission of electromagnetic energy through a waveguide is a solution to Maxwell's equations?
An infinitely thin precisely tuned dipole antenna has a pure resistance of 73 ohms. Why?
The graphs of sideband strengths of FM signals, as a function of modulation factor, are clearly the Bessel functions. Why? [I suspect this is way too advanced.]
How is the Laplace transform related to the Fourier transform? Why is it used? What does it do that the Fourier transform doesn't do? Why do the locations of poles and zeros dictate the behavior of filters and filter-like circuits?
What is the relationship between vectors (specifically orthogonal vectors) and the "phasing" method for generating single-sideband transmission? What does it mean to say that what comes out of the modulator is a vector?
What is the relationship between a linear operator and a matrix? Why do similar matrices denote identical operators? Or is it just "similar" operators?
What is the intuitive physical significance of an operator (or matrix?) being symmetric?
What is the intuitive physical significance of the eigenvalues and eigenvectors of an operator (or matrix)? What does it mean to say that the eigenvalues and eigenvectors completely characterize the operator?
What (if any) is the intuitive significance of the determinant of a matrix? Of an operator? Why does the standard procedure for calculating a determinant (pick any row or column, do stuff with cofactors) work? Please don't tell me about alternating tensors, or "universal maps", or "wedge products". It can't be that hard (or can it?) It's just a bunch of numbers!
Why does the product rule for determinants work?
What the heck is a tensor? [That's a common question, and I suspect the answer is simply out of these people's reach. And unnecessary for what they want to do. But they do ask.]
Maxwell's equations are said to be the only part of "classical" (pre-relativity) physics that didn't need to be rewritten for special relativity? What does that mean? And why is it so? Do the electric and magnetic field just transform correctly? Even at relativistic speeds?
As you can see, these kids are very inquisitive, and very smart! I have pointed them at Conservapedia, and told them to look around and see what they can learn.
I hope that a course in linear algebra will come into existence. There seems to be a good deal of interest in the topic. Better still, I hope that a course in applications to electronics will come into existence.
HarryJP 23:14, 26 May 2010 (EDT)
Update, June 2
The students and I discussed this yesterday, and the consensus is that we will not be using Conservapedia as a source. Some of the students were quite adamant and outspoken on this point. However, I may work on some lectures on electronics. I know very little about serious mathematics, and what I do know I would not presume to write about---you have a lot of good authors here. But I think I can write about electronics, and there seems to be virtually nothing here on the subject.
HarryJP 00:12, 3 June 2010 (EDT)