Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics (Fundamental Theories of Physics, 5) by D. Hestenes (PDF)

Description

Matrix algebra has been called “the arithmetic of higher mathematics” [Be]. We think the basis for a better arithmetic has long been available, but its versatility has hardly been appreciated, and it has not yet been integrated into the mainstream of mathematics. We refer to the system commonly called ‘Clifford Algebra’, though we prefer the name ‘Geometric Algebm’ suggested by Clifford himself. Many distinct algebraic systems have been adapted or developed to express geometric relations and describe geometric structures. Especially notable are those algebras which have been used for this purpose in physics, in particular, the system of complex numbers, the quatemions, matrix algebra, vector, tensor and spinor algebras and the algebra of differential forms. Each of these geometric algebras has some significant advantage over the others in certain applications, so no one of them provides an adequate algebraic structure for all purposes of geometry and physics. At the same time, the algebras overlap considerably, so they provide several different mathematical representations for individual geometrical or physical ideas.

User’s Reviews

Editorial Reviews: Review `… future authors will owe a great debt to Professors Hestenes and Sobczyk for this pioneering work.’ Foundations of Physics, 16, 1986 `I repeat that GC enriches and simplifies everything it touches, not just on an advanced level but also, and perhaps even more so, on an elementary level. I am convinced that GC should be taught to undergraduate in place of the traditional approaches to vector algebra and analysis.’ James S. Marsh in American Journal of Physics `If the physics community seizes the opportunity represented by this book, and I hope it does, this book will become the handbook and the bible of GC.’ American Journal of Physics, 53:5 (1985) About the Author David Hesteness is awarded the Oersted Medal for 2002. The Oersted Award recognizes notable contributions to the teaching of physics. It is the most prestigious award conferred by the American Association of Physics Teachers.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐This is a mathematician’s book, for (mostly) mathematicians. In a sense, it’s Hestenes’ magnum opus. It’s not for the newbie – moe like a College Outline Series, that presumes the reader has already been through the coursework and simply needs a “pull it all together” refresher. Others of Hestenes’ books present the material at sub-sonic rate (so to speak) and some of his British deciples (Gull, Doran, Lasenby – all at Cambridge) have produced stellar examples of pedagogy. They usually refer back to this volume. While organized from elementary to complex, it is no more a page-turner than is any encyclopedia. Thus, my 4 stars are for enjoyment – for usefulness as a Urim & Thummim to all the other Geometric Algebra books out there, it would get 5 stars.That said, there is already one bit of Geometric Algebra (the geometric product of two bivectors, in Minkowski space) for which I thought I had discovered an exception to the usually-published rules for grade-promotion/demotion in geometric products, and was scratching my head about possibly having done something wrong. I could find no reference to the matter – until I stumbled upon it on page 13 of THIS volume. My derivation was correct, and the matter only gets 1/2 column-inch. So – as information-rich and densely-written as any math/physics volume I’ve owned – since that College Outline Series on Calculus that I inadvisedly bought as a high school student, the summer before I took calculus in school.

⭐First a rant: What has happened to the once high art of bookbinding? I have hardcover books, with sewn-in signatures, that have stood up well to decades of extensive use. Some softcover books, particularly the older ones from Dover Publications, have also stood up well. But virtually the entire publishing industry seems to have given up on the idea of making well-constructed books for reasonable prices. “Clifford Algebra to Geometric Calculus”, with its poorly glued spine and paper cover, is my outstanding example of the _worst_ of this trend. Even though I handle my books with reasonable care, my not-inexpensive softcover of “Clifford Algebra to Geometric Calculus” started to fall apart almost immediately, before I even made it halfway through Chapter 1. I would have purchased the hardcover version with a “library binding”, but the extra expense (more than $150) was so extravagant as to be beyond my means. Another complaint is that the printing seems to be somewhat fuzzy. This is an important book which I want in my personal library, so I’ll buy another copy should publication rights ever go to Dover. But in the meantime, __0 stars__ for the physical construction and printing of the softcover edition.I believe I’ve seen most, if not all, of the limited number of books on geometric algebra and its calculus, and this book (the first) remains the most comprehensive, but not the most comprehensible. A tremendous amount of insightful thinking went into this book, and a diligent and thoughtful reader will learn a lot. But the book cannot be considered the place to begin one’s study of geometric algebra. Understanding what Hestenes and Sobczyk are saying and trying to see its significance is maddening. There’s a quotation, from whom I’ve forgotten, to the effect that “Pioneering work is clumsy.” _5 stars_ for the content, but only_2 stars_ for readability.There are several other books which one should go to first. Hestenes’ own “New Foundations for Classical Mechanics”, written on about a junior or senior level, is much more clearly written than was his older “Clifford Algebra to Geometric Calculus” (graduate level). Easiest to read by far (sophomore level) and offering lots of contact with traditional courses on linear algebra and vector calculus are Alan Macdonald’s inexpensive “Linear and Geometric Algebra” and its follow-up, “Vector and Geometric Calculus”; Macdonald’s books also introduce one to use of free geometric algebra software. Clear writing, geometric insight, and lots of informative figures characterize “Geometric Algebra for Computer Scientists”, by Dorst, Fontijne, and Mann; be warned that the book has an extremely lengthy errata list at its website, so the book should only be read with a printout nearby of that list. Students wanting to see applications to physics other than classical mechanics should consider “Geometric Algebra for Physicists” (graduate level), by Doran and Lasenby.As noted above, Dorst, Fontijne, & Mann maintain a website where errata for their book may be found, and so also do Doran & Lasenby and Macdonald for their books. That’s a big help for the puzzled student who can’t figure out why something doesn’t make sense. But if there’s an errata list for “Clifford Algebra to Geometric Calculus”, I haven’t been able to Google it.A cautionary note when comparing the various books named: Hestenes and Sobczyk use a kludgily defined “inner product” between elements of the geometric algebra. Hestenes was the pioneer in the field, so many subsequent writers have adopted his inner product, most notably Doran and Lasenby. But Dorst has argued convincingly for replacement of the Hestenes inner product by more-cleanly-defined, easier-to-use, and easier-to-interpret, left and right “contraction products”. Macdonald uses the left contraction product exclusively, although he calls it the inner product and denotes it by a centered dot instead of by Dorst’s right floor bracket.

⭐i have been working a few years in geometric calculus and i believe this book should be in every house of every geometrist and every person that is intersted in geometric concepts with physics applications

⭐The intended audience of this book is mathematicians who already know differential geometry. It is definitely not for beginners. It is terse and contains no figures to aid the reader’s intuition. Few motivations are given and no applications are discussed. I would have expected the author to make every effort to help a wider audience – also non-mathematicians – to become familiar with the important ideas contained in his book. But no – teaching is clearly not one of his strengths. Anyone who would like to learn geometric algebra and calculus on their own would be advised to start with other books, such as (1) Doran & Lasenby, “Geometric Algebra for Physicists”, (2) Alan MacDonald, “Linear and Geometric Algebra” and “Vector and Geometric Calculus”, (3) John Snygg, “Clifford Algebra – A Computational Tool for Physicists” and “A New Approach to Differential Geometry using Clifford’s Geometric Algebra”. But unfortunately also these books are either too brief, too terse, or lacking in other ways. A really good introduction to geometric algebra and calculus for physicists is still missing.

⭐This book is very dry and chock full of definitions but as a book it gives you no insights into the heart of geometric algebra, or as Tristan Needham in complex analysis say Math is like music we are encouraged to read music but not to play it, just as we are encouraged to look at math but not to visualize it, if you want books that give you visual explanations look for geometric algebra by leo dorst or geometric algebra by alan mcdonald.

⭐Well written, clear and a great source of information. This is one of the best book available about Clifford Algebra

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