PURE AND APPLIED MATHEMATICS
A Program of Monographs, Textbooks, and Lecture Notes
Vector and Tensor Analysis
Second Edition, Revised and Expanded
Eutiquio C. Young
Department of Mathematics Florida State University
Tallahassee, Florida
Library of Congress Cataloging-in-Publication Data
Young, Eutiquio C.
Vector and tensor analysis / Eutiquio C. Young -- 2nd ed., rev. and expanded.
p. cm. -- (Monographs and textbooks in pure and applied mathematics; 172)
Includes bibliographical references and index. ISBN 0-8247-8789-7 (alk. paper)
1. Vector analysis. 2. Calculus of tensors. I. Title. II. Series.
QA433.Y67 1992
515'.63--dc20 92-33741 CIP
This book is printed on acid-free paper.
Copyright © 1993 by MARCEL DEKKER, INC. All Rights Reserved.
Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.
MARCEL DEKKER, INC.
270 Madison Avenue, New York, New York 10016
Current printing (last digit): 10 9 8 7 6 5 4 3 2 1
Preface to the Second Edition
In this new edition we have tried to maintain the objective of the first edition, namely, to acquaint students with the fundamental concepts of vector and tensor analysis together with some of their physical applications and geometrical interpretations, and to enable students to attain some degree of proficiency in the manipulation and application of the mechanics and techniques of the subject. We have tried to retain the qualities and features of the previous edition, placing great emphasis on intuitive understanding and development of basic techniques and computational skills.
In this edition each chapter has been rewritten and certain chapters have been
reorganized. For example, in Chapter 3 the section on directional derivatives of vector fields has been deleted, the section on transformation of rectangular cartesian coordinate systems, together with the invariance of the gradient, divergence and the curl has been incorporated in the discussion of tensors. In Chapter 4 the section on test for
independence of path has been combined with the section on path independence. In each chapter we have expanded discussions and provided more examples and figures to demonstrate computational techniques as well as to help clarify concepts. Whenever it is helpful we have introduced subtitles in each section to alert students to discussion of new topics. Throughout the book, we have written statements of definitions and theorems in boldface letters for easy identification.
The author will appreciate receiving information about any errors or suggestions for the improvement of this book. The author also wishes to thank Miss Deirdre Griese,
Production Editor, and her staff for assistance rendered in the revision of this book.
Preface to the First Edition
This book is intended for an introductory course in vector and tensor analysis. In writing the book, the author's objective has been to acquaint the students with the various
fundamental concepts of vector and tensor analysis together with some of their
corresponding physical and geometric interpretations, as well as to enable the students to attain some degree of proficiency in the manipulation and application of the mechanics and techniques of the subject.
Throughout the book, we place great emphasis on intuitive understanding as well as geometric and physical illustrations. To help achieve this end, we have included a great number of examples drawn from the physical sciences, such as mechanics, fluid
dynamics, and electromagnetic theory, although prior knowledge of these subjects is not assumed. We stress the development of basic techniques and computational skills and deliberately de-emphasize highly complex proofs. Teaching experience at this level suggests that highly technical proofs of theorems are difficult for students and serve little purpose toward understanding the significance and implications of the theorems. Thus we have presented the classical integral theorems of Green, Gauss, and Stokes only intuitively and in the simplest geometric setting. At the end of practically every section, there are exercises of varying degree of difficulty to test students' comprehension of the subject matter presented and to make the students proficient in the basic computation and techniques of the subject.
As a prerequisite for a course based on this book, the students must be familiar with the usual topics covered in a traditional elementary calculus course. Specifically, the students must know the basic rules of differentiation and integration, such as the chain rule, integration by parts, and iterated integration of multiple integrals. Although a knowledge of matrix algebra would be helpful, this is not an essential prerequisite. The book
requires only the bare rudiments of this subject, and they are summarized in the text.
The author wishes to thank his colleagues Professor Steven L. Blumsack, Wolfgang Heil, David Lovelady, and Kenneth P. Yanosko for reviewing portions of the manuscript and offering valuable comments and suggestions, and Professors Chiu Yeung Chan and Christopher K. W. Tam for testing the material on tensors in their classes during the developmental stage of the book. Last but not least, the author acknowledges with gratitude the assistance rendered by the production and editorial department of the publisher.
Contents
Preface to the Second Edition v
Preface to the First Edition vii
Chapter 1
Vector Algebra 1
1.1 Introduction 1.2 Definition of a Vector 1.3 Geometric
Representation of a Vector 1.4 Addition and Scalar Multiplication 1.5 Some Applications in Geometry 1.6 Scalar Product 1.7 Vector Product 1.8 Lines and Planes in Space 1.9 Scalar and Vector Triple Products
Chapter 2
Differential Calculus of Vector Functions of One Variable 75
2.1 Vector Functions of a Real Variable 2.2 Algebra of Vector Functions 2.3 Limit, Continuity, and Derivatives 2.4 Space Curves and Tangent Vectors 2.5 Arc Length as a Parameter 2.6 Simple Geometry of Curves 2.7 Torsion and Frenet-Serret Formulas 2.8 Applications to Curvilinear Motions 2.9 Curvilinear Motion in Polar Coordinates 2.10 Cylindrical and Spherical Coordinates
Chapter 3
Vector Field 3.6 Curl of a Vector Field 3.7 Other Properties of the Divergence and the Curl 3.8 Curvilinear Coordinate Systems 3.9 Gradient, Divergence, and Curl in Orthogonal Curvilinear Coordinate Systems
Chapter 4
Integral Calculus of Scalar and Vector Fields 207
4.1 Line Integrals of Scalar Fields 4.2 Line Integrals of Vector Fields 4.3 Properties of Line Integrals 4.4 Line Integrals Independent of Path 4.5 Green's Theorem in the Plane 4.6 Parametric Representation of Surfaces 4.7 Surface Area 4.8 Surface Integrals 4.9 The
Divergence Theorem 4.10 Applications of the Divergence Theorem 4.11 Stokes' Theorem 4.12 Some Applications of Stokes' Theorem
Chapter 5
Tensors in Rectangular Cartesian Coordinate Systems 307
5.1 Introduction 5.2 Notation and Summation Convention 5.3 Transformations of Rectangular Cartesian Coordinate Systems 5.4 Transformation Law for Vectors 5.5 Cartesian Tensors 5.6 Stress Tensor 5.7 Algebra of Cartesian Tensors 5.8 Principal Axes of Second Order Tensors 5.9 Differentiation of Cartesian Tensor Fields 5.10 Strain Tensor
Chapter 6
Tensors in General Coordinates 373
6.1 Oblique Cartesian Coordinates 6.2 Reciprocal Basis; Transformations of Oblique Coordinate Systems 6.3 Tensors in Oblique Cartesian Coordinate Systems 6.4 Algebra of Tensors in Oblique Coordinates 6.5 The Metric Tensor 6.6 Transformations of
Solutions to Selected Problems
469
