Vladimir Lepetic - Classical Vector Algebra

Every physicist and engineer, and certainly every mathematician, would undoubtedly agree that vector algebra is one of the basic mathematical instruments in their toolbox.

Classical Vector Algebra should be viewed as a prerequisite for, and an introduction to, other mathematical courses dealing with vectors, and it follows the typical form and appropriate rigor of more advanced mathematics texts.

The vector algebra discussed in this book briefly addresses vectors in general 3-dimensional Euclidean space, and then, in more detail, looks at vectors in Cartesian 3 space. These vectors are easier to visualize and their operational techniques are relatively simple, but they are necessary for the study of Vector Analysis. In addition, this book could serve as a good way to build up intuitive knowledge for more abstract structures of -dimensional vector spaces.

Definitions, theorems, proofs, corollaries, examples, and so on are not useless formalism, even in an introductory treatise they are the way mathematical thinking has to be structured. In other words, an introduction and rigor are not mutually exclusive.

The material in this book is neither difficult nor easy. The text is a serious exposition of a part of mathematics that students need to master in order to be proficient in the field. In addition to the detailed outline of the theory, the book contains literally hundreds of corresponding examples and exercises.

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Vladimir Lepetic

First edition published 2023

by CRC Press

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CRC Press is an imprint of Taylor & Francis Group, LLC

2023 Vladimir Lepetic

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ISBN: 9781032381008 (hbk)

ISBN: 9781032380995 (pbk)

ISBN: 9781003343486 (ebk)

DOI: 10.1201/9781003343486

Typeset in Palatino

by Newgen Publishing UK

Every physicist and engineer, and certainly every mathematician, would undoubtedly agree that vector algebra is among the basic mathematical instruments in their toolbox.

The title Classical Vector Algebra might be misconstrued as something particular, or something different from simple vector algebra. That is not the case. The adjective classical, for lack of a better word, was used on purpose, for two reasons: first, in order to avoid the term simple which, arguably, is much disliked by students; second, to differentiate it from, say, Vector Calculus, Linear Algebra, or parts of Differential Geometry (which, of course, are separate fields on their own). In other words, the vector algebra discussed in this book briefly addresses vectors in general 3-dimensional Euclidean space, and then, in more detail, considers vectors in Cartesian R3 space. These vectors are easier to visualize, and their operational techniques are relatively simple, but they are necessary for the next step, which is the study of Vector Analysis. In addition, this book could serve as a good way to build up the intuition needed for more abstract structures of n-dimensional vector spaces.

Having said all that, the present book should be viewed as a prerequisite for, and an introduction to, other mathematical disciplines dealing with vectors, and it follows the typical form and appropriate rigor of more advanced math texts. Definitions, theorems, proofs, corollaries, examples, and so on are not useless formalism, even in an introductory treatise they are the way in which mathematical thinking has to be structured. In other words, the terms introduction and rigor do not exclude one another they should complement each other. This is not to say that the material in this book is difficult, nor that it is easy. It is simply an attempt to give a serious exposition of a part of mathematics that everybody working in the above-mentioned disciplines needs to master in order to be proficient in his/her field. In addition to the detailed outline of the theory, the book contains literally hundreds of corresponding examples and exercises. This author hopes that the reader will complete at least some of them.