Geometrical Vectors

Author: Gabriel Weinreich
Publisher: University of Chicago Press
Pages: 126
ISBN: 978-0226890487
Print: 0226890481
Kindle: B01EYG40HO
Audience: Mathematicians, physicists and engineers.
Rating: 5
Reviewer: Mike James
Geometrical Vectors - are there any other kind?

A more interesting question is how many types of geometrical vector are there? This book is all about expanding your ideas of vectors to well beyond the basic "magnitude and direction" definition.

I should say early on that this is not a book about Geometric Algebra - that is a different, but related, topic. This is a book about the naive understanding of what makes a vector a physical something. It is a book about mathematics and you need some math to understand it, but it isn't a traditional theorem proof-style of math book. What it is is a discussion of the ideas that motivate the math and as such it is suitable it you are going on to learn about almost any topic that is based on vectors - linear algebra, differential manifolds, geometric algebra and so on. If you are a programmer then you will get to discover what covariance and contravariance means in much more detail than you could ever possibly need. In other words, this is not a book you need to read unless you really want a deep understanding of what quantities that have "direction" in their makeup are all about.

The book starts off with some orientation - fitting for a book on vectors. It tries to make you step back from what you might already think you know about such quantities and explains the idea of a topological view of the space of vectors, i.e. no measurement.

The next two chapters start to build up a picture on the variation possible on something with direction built-in. Here we meet the co- and contra-variant vector although not in coordinate or matrix terms. If you already know some linear algebra you will have to work hard to lose your preconceptions or to make the connections with what you arlready know.

In Chapter 4, the idea of dot- and cross-products motivates much of the expansion of the vector types. By this point in the book you may well be reeling from this new collection of ideas. My advice is to let it sink in - take a break and re-read it after a week or so. It will serve you well for the future.

After these revelations the book moves on to consider fields and what you might think of as vector calculus, although you might well now be happy to call  it geometrical calculus. Here we meet the gradient, curl and div in their natural settings. Only in Chapter 6 do we move on to consider coordinates - so uncool - but so necessary.

Chapter 7 introduces "The Grand Algebraization Rule" which seems so counter-intuitive if you have been taught the generalizations of the vector calculus operators in different coordinate systems. The rule says that their form stays the same -  but your former experience probably leads you to believe that they don't. I won't spoil the surprise in discovering how it all works.

Chapter 8 moves us from topological spaces to metric spaces which is where most physics is done and we meet Maxwell's equations. Finally there is a chapter outlining where we need to go next and how some of the geometric imaginings break down.

This is a deep book that will take you a long time to read, even though it only has 120 pages - such is the value of mathematical textbooks. It attempts to fix the miss-learning we nearly all suffer from by learning about vectors in a Cartesian orthogonal frame where all types of vectors have the same components and are hopelessly confused. Separating them out is the job of this book and it does it really well. Everyone interested in physics should read this and nearly all mathematicians too.

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Last Updated ( Tuesday, 01 June 2021 )