Grassmann Algebra Volume 1: Foundations: Exploring extended vector algebra with Mathematica

The publication of this book, Grassmann Algebra by John Browne, is a watershed event. According to the author, "the focus of these books [a second volume on applications is in preparation] is to provide a readable account in modern notation of Grassmann's major algebraic contributions to mathematics and science." I believe that the author has succeeded in realizing his goal. The only background required is linear algebra.For those who have never heard of Grassmann Algebra, let me address this question: as a subject, is Hermann Grassmann's algebra worth studying? Gauging by its influence, I would say `most definitely'. For instance, the vector calculus used in physics and engineering is a tailoring (by Gibbs and Heaviside) of Grassmann's algebra ([...]). Geometric Algebra, used in physics, engineering and computer science, was developed by William Clifford and based on the work of Grassmann ([...]). The list goes on. Here is an interesting biography of Elie Cartan that refers to Hermann Grassmann ([...]).If Hermann Grassmann's work is so influential why haven't you heard of him or his work? Unfortunately, Grassmann's mathematical work was generally unappreciated in his life. To quote Wikipedia, "The Die Lineale Ausdehnungslehre (1844) by Hermann Grassmann was a revolutionary text, too far ahead of its time to be appreciated [...])." In the end, is seems that those who did understand Grassmann's work, had to adapt it into various forms that were more accessible to the larger mathematical community. The adaptations became the primary sources. This resulted in Grassmann's work being forgotten except by a few specialists. Recent interest in Geometric Algebra has had the side effect of increasing interest in Grassmann Algebra. However, until the publication of Grassmann Algebra by John Browne, there has been no "one-stop" resource that provides the full mathematical story of Grassmann Algebra.So what should you expect from the book "Grassmann Algebra" by John Browne? The story takes over 500 pages to tell. Along the way, there will likely be concepts that you know, such as point and vector, whose definitions will change in subtle ways from those that you've heard before. There will likely be ideas that will be totally new, e.g., the idea of multiplying a point and a vector. The chapters start with the simplest and work their way to more sophisticated notions. Throughout, you will likely see a very neat interplay between geometry and algebra. It is this interplay that forms a main theme and keeps one's attention and interest. Chapter by chapter the picture of the entire system becomes more complete. All the terms are clearly defined. The pace is very even. To facilitate learning, the book is integrated with a Mathematica package which I found helpful.I find it very curious that Grassmann, in the forward of his 1862 book, Die Ausdehnungslehre, states that "a time will come when it [the results of all his efforts] will be drawn forth from the dust of oblivion and the ideas laid down here will bear fruit..." I believe that the time that Grassmann was speaking about is now and that John Browne's book initiates the period where Grassmann's work will move from obscurity to its rightful place in the mathematical sciences.

Today, Grassman's Algebra has come into its own after languishing for too many years in obscurity. Grassman's Algebra is also known as "Algebraic Geometry" meaning that it's possible to work out geometric relationships by means of algebra. There are many new concepts in this algebra, especially those that involve dimension. It's very nice to have everything in one place when something like this emerges because it's a pleasure to have a readable and good reference book. Something that's not mentioned that's of an enormous benefit is a CD that is included that can be used in conjunction with Mathematica. This presents the beneficial interactive environment that's required for a great learning experience.

I thought that reading about Grassman Algebra would be a step down from reading about Geometric Algebra. But in some ways, this is more profound. The basics that distinguish points from vectors, and use the anti-wedge, are really enlightening. I'm looking forward to a Volume 2 that has applications.

It has been said that unless you can program the computer to calculate something, you don't really understand the concept behind such calculation. The author, having written Mathematica programs to calculate just about every conceivable Grassmann algebraic concept, is well-positioned to explain Grassmann algebra in clear details. I have read much of his book on-line a few years ago, but am glad to have my own copy now.

The best entertainment $30 will provide. The gift that keeps on giving.

A classic

If you don't have Mathematica you will probably hate this book and that is a shame because if you remove all the Mathematica stuff (about 50% of the book, perhaps more towards the end), it would be a truly great text book on this fascinating algebra. The first few chapters are more-or-less standalone and they are really good. The author's writing style is excellent. He introduces the exterior product and the little-known regressive product as well. It's got a good index but it's laid out like a spreadsheet! Weird. Great book if you've got Mathematica, look elsewhere if you haven't.

Verständliche, gut lesbarw Einführung zum Thema Grassmann Algebra. Kannte das Buchprojekt durch das Internet. War so interessant, daß ich das Buch haben wollte.