Geometric Algebra for Physicists

Rating:
Summary: Compared to what ?
Review: The quality and importance of this book could hardly be overstated. Geometric algebra might casually be considered the "correct" generalization of linear algebra. By considering, for a start, directed line segments, the linear algebra courses presently taught in some high schools and all universities achieve miracles. Although viewed by a few of the slower students as merely unpleasant bookkeeping systems, linear algebra derives its power from allowing algebraic manipulation of sophisticated aggregate objects, namely vectors. The benefits are not just computational, but stem more importantly from a more powerful and more unified, although slightly more abstract point of view than a student had before studying. Geometric algebra is all that and much more. By extending consideration from directed line segments to the inclusion of direct plane segments, directed elements of three space, etc., an extremely flexible and elegant mathematical tool arises. It allows a deeper, quicker, and more concise treatment of essentially all of modern differential geometry. Its applications throughout physics are at once simplifications of ordinary matrix treatments and occasions to allow much greater insight.

Geometric algebra is a great theory, one of highest importance. It will, undoubtedly, find a dominant place in our mathematics curriculum at the highest speed allowed by our educational systems (the highest speed being actually quite slow). This book is an especially good place to begin study. It starts from the most elementary principles, and exposes the material with very thoughtful, clear presentation. The economy and elegance of the geometric algebra itself allows this one substantial but not enormous book to reveal great insights into many branches of study, from differential geometry and its applications to gravity theory to quantum mechanics and classical mechanics.

If I had no books in my library, I would purchase a Bible. If I had only the Bible in my library, I would purchase this book next. I would certainly study this book in all detail before making a third purchase. My library already has several books in it. None of them will be read further until I finish every line, every exercise of this book. It's an important theory, and it is explained in a very useful and articulate way. This would, of course, be entirely expected if the authors were from Oxford University. Since they are only from Cambridge, we might not have expected as much, but we got it, nonetheless.

Rating:
Summary: Articulate Path to the Future
Review: The quality and importance of this book could hardly be overstated. Geometric algebra might casually be considered the "correct" generalization of linear algebra. By considering, for a start, directed line segments, the linear algebra courses presently taught in some high schools and all universities achieve miracles. Although viewed by a few of the slower students as merely unpleasant bookkeeping systems, linear algebra derives its power from allowing algebraic manipulation of sophisticated aggregate objects, namely vectors. The benefits are not just computational, but stem more importantly from a more powerful and more unified, although slightly more abstract point of view than a student had before studying. Geometric algebra is all that and much more. By extending consideration from directed line segments to the inclusion of direct plane segments, directed elements of three space, etc., an extremely flexible and elegant mathematical tool arises. It allows a deeper, quicker, and more concise treatment of essentially all of modern differential geometry. Its applications throughout physics are at once simplifications of ordinary matrix treatments and occasions to allow much greater insight.

Geometric algebra is a great theory, one of highest importance. It will, undoubtedly, find a dominant place in our mathematics curriculum at the highest speed allowed by our educational systems (the highest speed being actually quite slow). This book is an especially good place to begin study. It starts from the most elementary principles, and exposes the material with very thoughtful, clear presentation. The economy and elegance of the geometric algebra itself allows this one substantial but not enormous book to reveal great insights into many branches of study, from differential geometry and its applications to gravity theory to quantum mechanics and classical mechanics.

If I had no books in my library, I would purchase a Bible. If I had only the Bible in my library, I would purchase this book next. I would certainly study this book in all detail before making a third purchase. My library already has several books in it. None of them will be read further until I finish every line, every exercise of this book. It's an important theory, and it is explained in a very useful and articulate way. This would, of course, be entirely expected if the authors were from Oxford University. Since they are only from Cambridge, we might not have expected as much, but we got it, nonetheless.

Rating:
Summary: A powerful mathematical language for physics and engineering
Review: This is a well-written book on a very interesting and important subject: geometric algebra (GA) is a powerful and elegant mathematical language -- based on the works of Hamilton, Grassmann and Clifford -- that is especially well-suited for spacetime physics and several fields of engineering.

The authors adopt David Hestenes' viewpoint of a graded GA as a unified mathematical language that is coordinate-free, thereby stressing the fundamental role of geometric invariants in physics.

In fact, the elementary vector analysis -- which pervades almost all undergraduate (and even) graduate approaches to electrodynamics -- finds its roots in the misguided Gibbsian approach: Gibbs advocated abandoning Hamilton's quaternions and just work with scalar and cross products of vectors. However, the cross product has a major flaw: it only exists in three (or seven) dimensions -- if we require that (i) it should have just two factors, (ii) to be orthogonal to the factors, and (iii) to have length equal to the corresponding parallelogram.

Electrodynamics and relativistic physics, particularly, are elegantly presented through GA and otherwise cumbersome calculations may be circumvented in a simple and insightful way.

Mainstream physics and engineering cannot overlook GA anymore.

Rating:
Summary: Compared to what ?
Review: This is truly a great book for any one who is interested in not just physics, but physical reality. Although the ideas expressed therein have a long history and are by no means as uniquely those of its authors as were Albert Einstein's in his day, I believe that they will have comparable lasting value. Moreover the synthesis presented in this book, which builds pre-eminently on the work of Hestenes, is absolutely superb. Interested readers need not take my word for these claims, but are invited to prove it to themselves.

Although the above should be a sufficient review, my experience nevertheless indicates that it is a good idea to warn potentially enthusiastic readers against several common semantic misconceptions, lest they jump to conclusions which prevent them from ever taking that vital first step. Thus let it be clearly understood that Geometric Algebra is NOT:
(1) A replacement for linear/matrix/tensor algebra (on the contrary, it is a very nice complement to these formalisms).
(2) Identical, or even very close, to Emil Artin's earlier excellent book on bilinear forms with the title "Geometric Algebra".
(3) Another name for the enormous field "algebraic geometry" (it is indeed appropriate that the word stemming from "geometry" comes first in "geometric algebra").
(4) Just another reformulation of complex / quaternion / octonian analysis; for it connects all these purely algebraic objects, and many generalizations thereof, to Felix Klein's Erlangen Programme and Sophus Lie's theory of continuous groups.
(5) The ultimate theory of everything (although it probably will eventually be found to have something to do with it).

Geometric algebra IS a practical and natural (canonical) tool for formulating physical and mathematical problems in homogeneous spaces in a fully covariant fashion. But more importantly, you do not need to understand all those words in order to benefit from it, and this book is an excellent place for physicists of all stripes to start.