“What the best and wisest parent wants for his own child, that must the community want for all its children. Any other ideal for our schools is narrow and unlovely; acted upon it destroys our democracy." – John Dewey, 1900

So much of my career in high-tech dealt with harnessing the electromagnetic spectrum for various uses to include communications, telemetry, radar, navigation, tracking, machine vision, and precision measurement to name a few. Fortunate to have worked with great minds in signal processing, I spent over two decades in industries ranging from electronic warfare, satellite communications, cellular, GPS, surveying and mapping, fleet management and automatic vehicle location, telematics, E911, and location-based services.

Today, more so than ever in our history, each of us is bombarded with natural, and human-made, electromagnetic waves, most having their roots in the simple trigonometric function: sin x.

Enjoy this brief introductory video about the electromagnetic spectrum as we know it. There are seven follow-on videos, all produced by NASA, and no longer than five minutes or so, that go into more detail on different bands in the electromagnetic spectrum.

Fortunately, we do not need to explore the electromagnetic spectrum as I needed to in my fields and waves coursework via Maxwell’s equations.

Independent consultant and junior college adjunct instructor. Former secondary math teacher who taught math intervention, algebra 1, geometry, accelerated algebra 2, precalculus, honors precalculus, AP Calculus AB, and AP Statistics. Prior to teaching, I spent 25 years in high tech in engineering, marketing, sales and business development roles in the satellite communications, GPS, semiconductor, and wireless industries. I am awed by the potential in our nation's youth and I hope to instill in them the passion to improve our world at local, state, national, and global levels.

The antiquated symbolism in which the equations of electromagnetism continue to be expressed is a depressing example of historical inertia. My God, the exterior calculus goes all the way back to Volterra!

Maxwell combined and added to the historical works of Volterra, Faraday, Gauss, Ampere, and other giants of science, as this article nicely expands upon.

Can you expand on your point re: antiquated symbolism?

The traditional notation for differential calculus in three dimensions is a mess. It depends on an identification of different components of the exterior algebra in three dimensions because they happen to have the same dimensions. But these components should be carefully kept apart since they are not canonically isomorphic. It also confuses vectors with covectors. It happens that in three dimensions one can reduce everything to three dimensional vector spaces but this relies on isomorphisms dependent on the arbitrary choice of an inner product and an orientation.

The price of any computational advantage here is conceptual confusion. In dimension n where n is
greater than 3 it is no longer possible to reduce everything to n dimensional vector spaces.

Ah, I understand. I attempt to explain situations like this to my advanced students as examples where the “black & white” nature of mathematics starts to unravel…I’ve even alluded to it being 50 or more shades of grey…

What particularly annoys me about some of the above posters is that they seem to take a delight in producing mystification and confusion. The traditional notation of multivariable differential calculus in which Maxwell’s Equations are often written almost seems to have been deliberately designed to create the maximum obscurity and opacity. It’s no wonder that students sometimes approach it as an exercise in memorizing wallpaper.

How would you represent those four equations so they are more understandable? I might even post it for you in contrast to those above.

For me, the two black & white posters are a means of poking fun at a very difficult course, which I’m sure was the case for many of my EE brethren (and sistren).

The whole issue of memorizing versus understanding spans nearly all of mathematics, and is a byproduct of forcing the learning of fixed curricula into time-specific periods. This dilemma impacts nearly all students regardless of where they reside in the universe of mathematics.

There are many excellent works on multivariable calculus whose authors try to explain as clearly as possible what is going on. The difficulty of some works has more to do with the horrible traditional notation used than anything inherent in the subject.

Rather than celebrating the befuddlement induced by the traditional notation of nineteenth century vector analysis we should discard it into the ashcan of history as rapidly as possible.

Let me mention that I have the greatest respect for the 19th century scientists you mentioned in your comment. They had no idea what they were doing but that’s OK. They were pioneers and they weren’t supposed to know what they were doing. But we shouldn’t perpetuate their confused notation.

The antiquated symbolism in which the equations of electromagnetism continue to be expressed is a depressing example of historical inertia. My God, the exterior calculus goes all the way back to Volterra!

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I’m not sure I understand your point, Jim.

Maxwell combined and added to the historical works of Volterra, Faraday, Gauss, Ampere, and other giants of science, as this article nicely expands upon.

Can you expand on your point re: antiquated symbolism?

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The traditional notation for differential calculus in three dimensions is a mess. It depends on an identification of different components of the exterior algebra in three dimensions because they happen to have the same dimensions. But these components should be carefully kept apart since they are not canonically isomorphic. It also confuses vectors with covectors. It happens that in three dimensions one can reduce everything to three dimensional vector spaces but this relies on isomorphisms dependent on the arbitrary choice of an inner product and an orientation.

The price of any computational advantage here is conceptual confusion. In dimension n where n is

greater than 3 it is no longer possible to reduce everything to n dimensional vector spaces.

LikeLike

Ah, I understand. I attempt to explain situations like this to my advanced students as examples where the “black & white” nature of mathematics starts to unravel…I’ve even alluded to it being 50 or more shades of grey…

LikeLike

What particularly annoys me about some of the above posters is that they seem to take a delight in producing mystification and confusion. The traditional notation of multivariable differential calculus in which Maxwell’s Equations are often written almost seems to have been deliberately designed to create the maximum obscurity and opacity. It’s no wonder that students sometimes approach it as an exercise in memorizing wallpaper.

LikeLike

I meant to say “in memorizing wallpaper patterns”. I,m upset.

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Sorry to hear you’re upset, Jim.

How would you represent those four equations so they are more understandable? I might even post it for you in contrast to those above.

For me, the two black & white posters are a means of poking fun at a very difficult course, which I’m sure was the case for many of my EE brethren (and sistren).

The whole issue of memorizing versus understanding spans nearly all of mathematics, and is a byproduct of forcing the learning of fixed curricula into time-specific periods. This dilemma impacts nearly all students regardless of where they reside in the universe of mathematics.

LikeLike

There are many excellent works on multivariable calculus whose authors try to explain as clearly as possible what is going on. The difficulty of some works has more to do with the horrible traditional notation used than anything inherent in the subject.

LikeLike

Rather than celebrating the befuddlement induced by the traditional notation of nineteenth century vector analysis we should discard it into the ashcan of history as rapidly as possible.

LikeLike

Let me mention that I have the greatest respect for the 19th century scientists you mentioned in your comment. They had no idea what they were doing but that’s OK. They were pioneers and they weren’t supposed to know what they were doing. But we shouldn’t perpetuate their confused notation.

LikeLike