Electrodynamics: A Modern Geometric Approach (Progress in Mathematical Physics) [Hardcover]

William E. Baylis (Author)

Book Description

ISBN-10: 0817640258 | ISBN-13: 978-0817640255 | Publication Date: December 23, 1998 | Edition: Corrected

The emphasis in this text is on classical electromagnetic theory and electrodynamics, that is, dynamical solutions to the Lorentz-force and Maxwell's equations. The natural appearance of the Minkowski spacetime metric in the paravector space of Clifford's geometric algebra is used to formulate a covariant treatment in special relativity that seamlessly connects spacetime concepts to the spatial vector treatments common in undergraduate texts. Baylis' geometrical interpretation, using such powerful tools as spinors and projectors, essentially allows a component-free notation and avoids the clutter of indices required in tensorial treatments. The exposition is clear and progresses systematically---from a discussion of electromagnetic units and an explanation of how the SI system can be readily converted to the Gaussian or natural Heaviside-- Lorentz systems, to an introduction of geometric algebra and the paravector model of spacetime, and finally, special relativity. Other topics include Maxwell's equation(s), the Lorentz-force law, the Fresnel equations, electromagnetic waves and polarization, wave guides, radiation from accelerating charges and time-dependent currents, the Liénard--Wiechert potentials, and radiation reaction, all of which benefit from the modern relativistic approach. Numerous worked examples and exercises dispersed throughout the text help the reader understand new concepts and facilitate self-study of the material. Each chapter concludes with a set of problems, many with answers. Complete solutions are also available. An excellent feature is the integration of Maple into the text, thereby facilitating difficult calculations. The text is designed for upper-level undergraduate and beginning graduate courses in mathematical physics or physics. It should also be

Editorial Reviews

Review

"[The book] has a nice blend of mathematical physics, fundamentals of electromagnetic theory and practical applications.... It is extremely well written and contains numerous exercises and problems to help the reader gain familiarity with new concepts. Throughout...the author emphasizes the conceptual framework of electromagnetism so that the reader does not get lost in details. Such a style is very valuable in a physics textbook.... The book simultaneously teaches the reader electromagnetic theory and more advanced mathematical concepts in a very concrete physical context.... Essentially self-contained...it will serve as an excellent textbook for graduate level physics courses on electromagnetic theory and mathematical physics. It is also a good reference book for researchers in applied mathematics, theoretical physics and electrical engineering."

—Current Science

Product Details

• Hardcover: 404 pages
• Publisher: Birkhäuser Boston; Corrected edition (December 23, 1998)
• Language: English
• ISBN-10: 0817640258
• ISBN-13: 978-0817640255
• Product Dimensions: 9.6 x 7 x 1 inches
• Shipping Weight: 1.9 pounds (View shipping rates and policies)
• Average Customer Review: 2.4 out of 5 stars  See all reviews (5 customer reviews)
• Amazon Best Sellers Rank: #449,536 in Books (See Top 100 in Books)

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86 of 87 people found the following review helpful:

5.0 out of 5 stars It's about time!, January 19, 2000

By 

kaiser (Austin, TX USA) - See all my reviews
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This review is from: Electrodynamics: A Modern Geometric Approach (Progress in Mathematical Physics) (Hardcover)

Maxwell's theory of Electrodynamics is considerably more complicated than Newton's Mechanics. The latter deals with concrete objects (particles, rigid bodies, etc.) while the former deals with intangible "fields" distributed in space, a concept that took many years to evolve and gain acceptance. It is therefore not surprising that Electrodynamics has motivated a variety of important scientific developments, designed either to simplify it conceptually or to make it consistent with Mechanics. In physics, fundamental contradictions between Electrodynamics and Mechanics spurred Einstein to develop Special Relativity. In mathematics, the most well-known development is vector analysis, introduced by Gibbs to simplify Maxwell's equations. Unfortunately, the even deeper simplifications introduced by Hamilton (based on quaternions) and Clifford (based on Clifford algebra) have not gained wide acceptance because they are somewhat more technically demanding. Thus almost all physics and engineering textbooks on electrodynamics use vector analysis, and very few students and researchers are even aware of the tremendous power offered by quaternionic and Clifford analysis.

In this regard, the book by Baylis is a real blessing. By choosing the simplest geometric algebra (complex quaternions, known in quantum mechanics as the Pauli algebra), Baylis manages to use the full power of Clifford algebra without giving up the familiar notation of vector analysis (gradient, curl, divergence). The only new ingredient is the associative product of two vectors, which unifies their inner product and their cross product and leads to tremendous new possibilities. Students and workers in the field will love the resulting beauty and simplicity, and especially the continuity of notation and concepts with the mainstream literature. Although the Pauli algebra is usually associated with nonrelativistic quantum mechanics, Baylis amply demonstrates that it is equally able to handle relativistic aspects (such as the unified electromagnetic field and Lorentz transformations) when the concept of three-dimensional vectors is replaced by four-dimensional (space-time) "paravectors." Well done!

16 of 22 people found the following review helpful:

2.0 out of 5 stars An awkward geometric approach, July 31, 2004

By 

Prof C. R. PAIVA "Carlos Paiva" (Lisbon, Portugal) - See all my reviews
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This review is from: Electrodynamics: A Modern Geometric Approach (Progress in Mathematical Physics) (Hardcover)

This book on electrodynamics presents an awkward geometric approach.

Indeed, electromagnetic theory is inherently relativistic -- although it is usually introduced in a nonrelativistic formalism that uses the elementary Gibbsian approach to vectors.

The author intends to avoid the tensor notation -- because it tends to disguise the corresponding geometrical significance. But he also intends to avoid spaces of nondefinite metric by using paravectors in a Clifford algebra with dimension n=3. In fact, in this process, he needs to introduce a complex structure where trivectors become imaginary scalars, bivectors are written as imaginary vectors, and every element of the algebra is a paravector -- the sum of a scalar and a vector, both of which may be complex.

One should recognize that the author has grasped the problem; his solution is, nevertheless, foggy. All in all, one cannot say -- quoting Arnold Sommerfeld -- that "the true mathematical structure of these entities will appear only now, as in a mountain landscape when the fog lifts".

Much ado about nothing: I recommend, instead, other geometric algebras that don't shy away from using indefinite metrics -- but avoid the clumsiness of complex structures.

Indeed, the real algebra of complex numbers is already an example of a Clifford algebra: "the geometric view of complex numbers is connected with the structure of C as a real algebra, and not so much as a field" (Pertti Lounesto).

David Hestenes shows how this can be done.

2 of 2 people found the following review helpful:

2.0 out of 5 stars a curious melange of material, not entirely effective., November 13, 2010

By 

Peeter Joot "Peeter Joot" (Markham, Ontario, Canada) - See all my reviews
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This review is from: Electrodynamics: A Modern Geometric Approach (Progress in Mathematical Physics) (Hardcover)

I quite liked the geometric algebra introduction in this book. It was one of the simplest and clearest I've read. That is, until the paravector notation is introduced. I felt overwhelmed by the variety of conjugation operations suddenly tossed out (clifford conjugation, hermitian conjugation, grade automorphism).

Having learned what I know of Geometric Algebra based E&M with the STA formalism, this approach seems more difficult. The complexity (literally) of having to deal with conjugation and scalar and vector selection seems much less intuitive.

Some of the tail end chapters almost seemed tossed in randomly. It was hard to see what value GA provided in the spherical harmonic treatment.

I'd hoped that this text would provide an easy path to learn E&M in a GA context despite a requirement for having to switch from STA notation to APS, but didn't feel the presentation provided a logical sequential flow of ideas that would facilitate this study.