Geometrical Vectors

By Gabriel Weinreich

Every advanced undergraduate and graduate student of physics must master the concepts of vectors and vector analysis. Yet most books cover this topic by merely repeating the introductory-level treatment based on a limited algebraic or analytic view of the subject.

Geometrical Vectors introduces a more sophisticated approach, which not only brings together many loose ends of the traditional treatment, but also leads directly into the practical use of vectors in general curvilinear coordinates by carefully separating those relationships which are topologically invariant from those which are not. Based on the essentially geometric nature of the subject, this approach builds consistently on students' prior knowledge and geometrical intuition.

Written in an informal and personal style, Geometrical Vectors provides a handy guide for any student of vector analysis. Clear, carefully constructed line drawings illustrate key points in the text, and problem sets as well as physical examples are provided.

• Language: English
• Publisher: University of Chicago Press
• Release date: Jun 5, 2020
• ISBN: 9780226778693

Book preview

Index

PREFACE

Years of teaching Mathematical Methods of Physics at the University of Michigan to seniors and first-year graduate students convinced me that existing textbooks don’t do an adequate job in the area of vector analysis: all too often, their treatment is a repetition of what students had already seen in earlier courses, with little or no insight into the essentially geometrical structure of the subject. For this reason, I got into the habit of substituting my own discussion for whatever the textbook contained, a few years ago even going so far as to distribute some informal notes under the title Geometrical Vectors.

Reactions to those notes, on the part of both colleagues and students, were enthusiastic, and the thought of rewriting the material in a form suitable for publication naturally followed. It has, however, been clear to me from the beginning that the resulting book can only have an appreciable market if it is priced so that students can buy it in addition to one of the standard textbooks. Conversations with experts indicated that such a goal was, indeed, attainable in a paperback produced from my camera-ready copy, and so I got to work; three years later, the result of that work is before you.

In planning the book, I had to decide whether to call it Geometrical Vectors and limit its coverage correspondingly, or to go for Geometrical Vectors and Tensors, discuss both subjects, and in doing so double its size. If in fact I opted for the first, it was not without a great deal of sadness, because tensor analysis is so beautiful, while at the same time its elementary treatments tend to be no better than those of vector analysis. Yet the practical demand for physicists to understand tensors is quite small compared to the ubiquitous use of vectors; and I was afraid that the consequent increase in price might make the resulting book inaccessible to what is, as I see it, its primary audience.

My enormous indebtedness in this work is to so many that it can only be acknowledged generically. And so my deep and heartfelt appreciation goes, first, to my students for constantly asking questions; second, to my teachers for teaching me to do the same; and last but by no means least, to my many colleagues (including a number of anonymous reviewers) in the crucible of whose conversation my present understanding was refined. I sincerely hope that others, too, may now find this understanding useful.

Gabriel Weinreich

Ann Arbor

December 1997

1

PROLOGUE: WHAT THIS BOOK IS ABOUT

1.1 Introduction

We live in a three-dimensional flat space, or at least that is the way we usually think about it. It’s true that special relativity motivates us to add time as a fourth dimension, and general relativity to consider our space as curved, that is, not subject to the constraint of having the angles of every triangle necessarily adding up to 180°. Nonetheless, the normal, three-dimensional Euclidean space remains the space in which our intuition lives and breathes and has its being; it is, simply said, the only space that our mind can truly picture.

Because of this fact, it makes sense to concentrate our exploration of vectors in such a space, especially as there are some features of three-dimensional vectors – for example, the cross product – which have no exact equivalent in spaces of other dimensionality; we refer here to aspects that have nothing to do with any human ability to perceive or imagine, but arise out of the mathematical nature of the space. (This does not mean, of course, that such features cannot be generalized to two, or four, or N dimensions, but such a generalization always carries a price, in that at least some of the properties of the concept in question have to be abandoned.) Nonetheless, the reader will find that, even though our treatment concentrates on the intuitively familiar three-dimensional flat Euclidean space, it will still provide a sufficiently solid grounding to begin generalizing to other types of spaces (see Chapter 9).

1.2 Where We Begin

In our discussions, we will repeatedly refer to the traditional treatment of vector analysis, meaning one in which vectors are always represented by arrows, and vector operations, although they may have a geometrical embodiment, are usually defined in terms of algebraic (or differential) operations on Cartesian components. Although it is our assumption that the reader has encountered such a treatment before, not only in a first-year physics course but perhaps even in one or more intermediate courses, we summarize its essential procedures and formulas before going any further.

Addition: Geometrically, two vectors A and B are added by the parallelogram rule, according to which the tails of the two arrows are brought together, their parallelogram is completed, and a new vector C (which is the sum of A and B) is drawn from the common tail to the opposite vertex. Algebraically, the components of C are obtained by adding corresponding components of A and B:

Multiplication by a scalar: The length of the arrow is multiplied by the scalar, reversing direction if the scalar is negative. Algebraically, each component is multiplied by the scalar.

Scalar product of two vectors (dot product): The lengths of the two arrows are multiplied together, then further multiplied by the cosine of the angle between them. In terms of