Vector and Geometric Calculus (Geometric Algebra & Calculus)

[Readers of this review should note that I examined early drafts of this book and provided comments to the author. I have and have had no financial interest in it, other than to encourage the author to keep its price low.] This textbook covers the basic concepts of differential calculus as they occur in a multi-dimensional setting. Textbooks on this essential math and physics undergraduate topic have traditionally been entitled "Advanced Calculus," "Multivariate Calculus," and "Vector Analysis." The latter title refers to the specific study of scalar and vector fields on vector spaces, from which, for example, arise the differential operator concepts of curl and divergence that are so critical to the understanding of electromagnetism. There are numerous textbooks to choose from in this area. This one distinguishes itself by two attributes: its thoroughgoing use of Geometric Algebra and the clarity of its exposition at an undergraduate level. Some recent texts have provided clearer treatments of multivariate calculus by using coordinate-free algebraic notions from modern treatments of Linear Algebra. This text completes the trend by basing its exposition on the use of algebraic notions from Geometric Algebra, which provides a greatly enhanced command of linear and multilinear concepts. The author assumes the reader is familiar with this topic, and has already written a concise introduction to it in a separate textbook. Furthermore, and most significantly, this text employs the differential operator central to the development of the calculus portion of Geometric Algebra, the vector derivative. The use of this operator results in a generalized and simplified statement of Stoke's Theorem, the fundamental theorem of multivariate calculus. The style of the exposition is clearly suited to someone trying to learn this topic at the undergraduate level. It's attention to concept development and even pacing are especially well-suited to self-study. Professor MacDonald, as he has previously demonstrated in his textbook on Linear and Geometric Algebra, definitely has a gift for fitting a clear exposition of a new area into the framework of a traditional subject. I have great hopes that these textbooks will accelerate the long overdue introduction of Geometric Algebra to the undergraduate curriculum.

I have really enjoyed this book (and its prequel Linear and Geometric Algebra, which I would recommend reading first). I wish I could have learned and taken advantage of Geometric Algebra back when I was learning vector calculus a few decades ago. I find the approach offered in this book using Geometric Algebra is much more personally satisfying than Differential Forms. I like the way that Professor Macdonald combines intuition and practical application with a mathematical treatment. I highly recommend this book.

Geometric algebra is the most powerful tool to understand linear algebra, vector calculus, special and general relativity, and quantum physics. It is simple and intuitive.

It's taken far too long for David Hestenes' work on Geometric Algebra to catch on. Hopefully Alan Macdonald's excellent text, Vector and Geometric Calculus, will contribute to changing this. There are plenty of online tutorials, including Hestenes' Space-Time Algebra, but serious study should probably start with Doran and Lasenby's Geometric Algebra for Physicists. The Classic work is Clifford Algebra to Geometric Calculus by Hestenes and Sobcyzk, but I find Macdonald's book far more readable. By leaving much of the work as exercises, he covers a great deal of ground in a relatively small book. The book assumes a familiarity with his Linear Algebra and Geometric Algebra, which is also an excellent book, but Vector and Geometric Calculus is simply in a class by itself (as is the Doran and Lasenby book). To round things out, look at Hestenes New Foundations in Classical Mechanics. In short, if one were to read two books on Geometric Algebra and Calculus, I would start with Doran and Lasenby and then study Alan Macdonald's book. This is a topic whose time has come.

It is a perfect work. It is a long long way to understand GA, but it is worth trying. Where the Grassmann-Clifford-Hestenes GA denial of classical algebra will win the full scientific world view will be totally rewritten and a GA based unfication would happen giving the older dislocated, isolated otherwise beatiful results a general cant.

I just got the delivery from DHL transfered from Hongkong. After overviewing the book, it is an excellent extension of the calculus course in your college standard curriclumn. I would suggest reading Linear and Geometric Algebra frist. Because the concept would be understood easier.

Superb.

It's possible this book would be ok in a structured classroom setting. But, for self-study, the theorem/proof style unfortunately just doesn't work when many of the most important theorems are left as exercises for the reader to prove (particularly because there is no accompanying solutions manual).

This book is definitely a bit harder to read than "Linear and Geometric Algebra" (LAGA) but at the same time it is a very good and natural complement for that book. Having said that, I strongly recommend to any potential reader of this book to read first the LAGA book, as the author says in the introduction. The writing style is very similar to LAGA and the compactness of the definitions and introductions are still very appealing. However, as the author says in the introduction of LAGA, the reader should feel comfortable with the "theorem-proof" style of writing. I'm learning by myself the subject (I'm still at the middle of this book) and I should say that it is not an easy subject, but the beauty of geometric algebra and calculus definitely pays off. The way in which the author merges the "new" concepts of geometric calculus with the "old" concepts of vector calculus is very appealing. It makes feel to the reader that this is the way that the subject should have been learned (and taught) since the beginning (say in college or at undergraduate level). In general, my opinion with respect to this book is similar to LAGA. I like the compactness of the concepts, which in turns provides a good aid at grasping the actual subject and at the same time it is a very good reference for a quick review. I know it is a matter of taste, but I still think that the answers to some selected exercises or problems would be very helpful for some of us who are learning the subject without the help of an instructor. Looking at the interesting approach of the author at teaching this subject, I definitely recommend this book (and LAGA, of course) UPDATE: After finishing most of the book, I must say that I ended up loving it. In section 3 (integrals) there are some exercises with solutions, which result very helpful (just what I was asking for :-) ) . The chapter about the fundamental theorem of calculus is very well written and it is definitely THE reference for geometric calculus for everybody wanting to learn about these beautiful tools. Highly recommendable.

If you ever struggled to understand Stokes’ theorem and are interested in geometric or Clifford algebra and calculus, this is a great book to read. It is easy to read and systematic. As a topic geometric calculus is unfairly overlooked, as it is elegant and does not feel as hoc, the way that much of modern mathematics feels.

Mon mari semble ravi de ce livre et de son suivant !

Un'introduzione semplice ma efficace al linguaggio matematico che permette a mio parere la più grande unificazione, per altro intuitiva, dell'analisi.

Good book , well written but.....not enough examples for me..