The observation that has been made that many Geometric Algebra based
textbooks (e.g. Hestenes's classical mechanics book and Baylis's
electrodynamics) seem to begin by introducing GA *de novo*. Imagine
what a barrier to advancing physics students' programs would be
present if a thorough introduction to vector algebra and complex
algebra had to preface every textbook in the curriculum.
The proper place to introduce GA is not in undergrad or grad school.
David Hestenes believes it should be introduced earlier, before one
ever hears of the Gibbs (ordinary) vector algebra or complex numbers.
GA subsumes those formalisms very elegantly, and quaternions and
Pauli spin matrices *en passant*. "And more" as they say on TV!
John hasn't mentioned the more exciting aspects of GA. Have you ever
wanted to solve a vector equation by dividing both sides of it by a
vector? Well, that is a natural operation in GA.
I have a good friend who got his PhD as a student of Steve Gull's at
Cambridge. He did everything in GA; it had become his first language
of mathematics. The elegance John notes in the expression of
Maxwell's equation of electrodynamics is attainable in other ways, of
course, notably in special relativistic formalism. Once one sees GA,
however, it is difficult to imagine doing things any other way.
I'm reading a newly edited edition of Clifford's opus posthumous "The
Common Sense of the Exact Sciences" now. This 1955 Dover reprint of a
1946 book begins with a brief biography of William Kingdon Clifford.
I'll let you all know if it's worth reading after I finish it. It's a
bit tedious going in the first two chapters. Clifford didn't finish
those himself, and the editor's comments are not always reassuring.
(I am an enthusiast, something I was pushing on phys-l back in 1995.)