SUMS05 Vector Calculus, Paul C. Matthews (1998) .pdf

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Front Matter

Cover

Springer Undergraduate Mathematics Series

Advisory Board & List of Publist Books

Vector Calculus

ISBN 3540761802

QA433.M38 1998 515'.63-dc21

LCCN 97-41191

Second Indian Reprint 2008

ISBN 978-81-8128-295-8

Preface

Table of Contents

1 Vector Algebra

1.1 Vectors and scalars

1.1.1 Definition of a vector and a scalar

1.1.2 Addition of vectors

1.1.3 Components of a vector

1.2 Dot product

1.2.1 Applications of the dot product

EXERCISES

1.3 Cross product

1.3.1 Applications of the cross product

1.4 Scalar triple product

1.5 Vector triple product

1.6 Scalar fields and vector fields

Summary of Chapter 1

EXERCISES

2 Line, Surface and Volume Integrals

2.1 Applications and methods of integration

2.1.1 Examples of the use of integration

2.1.2 Integration by substitution

2.1.3 Integration by parts

2.2 Line integrals

2.2.1 Introductory example: work done against a force

2.2.2 Evaluation of line integrals

2.2.3 Conservative vector fields

2.2.4 Other forms of line integrals

EXERCISES

2.3 Surface integrals

2.3.1 Introductory example: flow through a pipe

2.3.2 Evaluation of surface integrals

2.3.3 Other forms of surface integrals

2.4 Volume integrals

2.4.1 Introductory example: mass of an object with variable density

2.4.2 Evaluation of volume integrals

Summary of Chapter 2

EXERCISES

3 Gradient, Divergence and Curl

3.1 Partial differentiation and Taylor series

3.1.1 Partial differentiation

3.1.2 Taylor series in more than one variable

3.2 Gradient of a scalar field

3.2.1 Gradients, conservative fields and potentials

3.2.2 Physical applications of the gradient

EXERCISES

3.3 Divergence of a vector field

3.3.1 Physical interpretation of divergence

3.3.2 Laplacian of a scalar field

3.4 Curl of a vector field

3.4.1 Physical interpretation of curl

3.4.2 Relation between curl and rotation

3.4.3 Curl and conservative vector fields

Summary of Chapter 3

EXERCISES

4 Suffix Notation and its Applications

4.1 Introduction to suffix notation

4.2 The Kronecker delta \delta_ij

4.3 The alternating tensor \epsilon_ijk

4.4 Relation between \epsilon_ijk and \delta_ij

EXERCISES

4.5 Grad, div and curl in suffix notation

4.6 Combinations of grad, div and curl

4.7 Grad, div and curl applied to products of functions

Summary of Chapter 4

EXERCISES

5 Integral Theorems

5.1 Divergence theorem

5.1.1 Conservation of mass for a fluid

5.1.2 Applications of the divergence theorem

5.1.3 Related theorems linking surface and volume integrals

EXERCISES

5.2 Stokes's theorem

5.2.1 Applications of Stokes's theorem

5.2.2 Related theorems linking line and surface integrals

Summary of Chapter 5

EXERCISES

6 Curvilinear Coordinates

6.1 Orthogonal curvilinear coordinates

6.2 Grad, div and curl in orthogonal curvilinear coordinate systems

6.2.1 Gradient

6.2.2 Divergence

6.2.3 Curl

EXERCISES

6.3 Cylindrical polar coordinates

6.4 Spherical polar coordinates

Summary of Chapter 6

EXERCISES

7 Cartesian Tensors

7.1 Coordinate transformations

7.2 Vectors and scalars

7.3 Tensors

7.3.1 The quotient rule

EXERCISES

7.3.2 Symmetric and anti-symmetric tensors

7.3.3 Isotropic tensors

7.4 Physical examples of tensors

7.4.1 Ohm's law

7.4.2 The inertia tensor

Summary of Chapter 7

EXERCISES

8 Applications of Vector Calculus

8.1 Heat transfer

8.2 Electromagnetism

8.2.1 Electrostatics

8.2.2 Electromagnetic waves in a vacuum

EXERCISES

8.3 Continuum mechanics and the stress tensor

8.4 Solid mechanics

8.5 Fluid mechanics

8.5.1 Equation of motion for a fluid

8.5.2 The vorticity equation

8.5.3 Bernoulli's equation

Summary of Chapter 8

EXERCISES

Solutions

Solutions to Exercises for Chapter 1

Solutions to Exercises for Chapter 2

Solutions to Exercises for Chapter 3

Solutions to Exercises for Chapter 4

Solutions to Exercises for Chapter 5

Solutions to Exercises for Chapter 6

Solutions to Exercises for Chapter 7

Solutions to Exercises for Chapter 8

Back Matter

Index

Back Cover

P.C. Matthews i U COW a moo A It=1) I I TV 11 I G P x.'(g)x,) Springer IT SPRINGER UNDERGRADUATE D MATHEMATICS S SERIES

Springer Undergraduate Mathematics Series

Advisory Board P.J. Cameron Queen Mary and Westfield College M.A.J. Chaplain University of Dundee K. Erdmann Oxford University L.C.G. Rogers University of Cambridge E. Siili Oxford University J.F. Toland University of Bath Other books in this series A First Course in Discrete Mathematics L Anderson Analytic Methods for Partial Differential Equations G. Evans, J. Blackledge, P. Yardley Applied Geometry for Computer Graphics and CAD, Second Edition D. Marsh Basic Linear Algebra, Second Edition T.S. Blyth and E.F. Robertson Basic Stochastic Processes Z Brzeiniak and T. Zastawniak Complex Analysis J.M Howie Elementary Differential Geometry A. Pressley Elementary Number Theory G.A. Jones and JM Jones Elements of Abstract Analysis M 6 Searcdid Elements of Logic via Numbers and Sets D.L. Johnson Essential Mathematical Biology N.F. Britton Fields, Flows and Waves: An introduction to Continuum Models D.F. Parker Further Linear Algebra T.S. Blyth and E.F. Robertson Geometry R. Fenn Groups, Rings and Fields D.A.R. Wallace Hyperbolic Geometry J.W. Anderson Information and Coding Theory G.A. Jones and J.M. Jones Introduction to Laplace Transforms and Fourier Series P.P.G. Dyke Introduction to Ring Theory P.M Cohn Introductory Mathematics: Algebra and Analysis G. Smith Linear Functional Analysis B.P. Rynne and M.A. Youngson Mathematics for Finance: An Introduction to Financial Engineering M. Capii ksi and T. Zastawniak Matrix Groups: An Introduction to Lie Group Theory A. Baker Measure, Integral and Probability, Second Edition M. Capiriksi and E. Kopp Multivariate Calculus and Geometry, Second Edition S. Dineen Numerical Methods for Partial Differential Equations G. Evans, J. Blackledge, P. Yardley Probability Models J.Haigh Real Analysis J.M. Howie Sets, Logic and Categories P. Cameron Special Relativity N.M.I. Woodhouse Symmetries D.L. Johnson Topics in Group Theory G. Smith and O. Tabachnikova Vector Calculus P.C Matthews

P.C. Matthews Vector Calculus With 63 Figures 1 Springer

Paul C. Mathews, PhD School of Mathematical Sciences, University of Nottingham, University'Park, Nottingham, NG7 2RD, UK Cover illustration elements reproduced by kind permission of Aptech Systems, Inc., Publishers of the GAUSS Mathematical and Statistical System, 23804 S.E. Kent-Kangley Road, Maple Valley, W A 98038, USA. Tel.: (206) 432-7855 Fax (206) 432-7832 email:info@aptech.com URL:www.aptech.com American Statistical Association: Chance Vol 8 No 1, 1995 article by KS and KW Heiner `Tree Rings of the Northern Shawangunks' page 32 fig 2 Springer-Verlag: Mathematica in Education and Research Vol 4 Issue 3 1995 article by Roman E Maeder, Beatrice Amrhein and Oliver Gloor'Illustrated Mathematics: Visualization of Mathematical Objects" page!) fig 11, originally published as a CD ROM 'Illustrated Mathematics'by TELOS: ISBN 0-387-14222-3, german edition by Birkhausen ISBN 3-7643-5100-4. Mathematica in Education and Research Vol 4 Issue 31995 article by Richard J Gaylord and Kazume Nishidate "Traffic Engineering with Cellular Automata" page 35 fig 2. Mathematics in Education and Research Vol 5 Issue 21996 article by Michael Trott'The Implicitization of a Trefoil Knot'page 14. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Lee de Cola 'Coins, Trees, Bars and Bells: Simulation of the Binomial Process' page 19 fig 3. Mathematica in Education and Research Vol 5 Issue 2 1996 article by Richard Gaylord and Kazume Nishida te 'Contagious Spreading' page 33 fig 1. Mathematica in Education and Research Vol 5 Issue 21996 article by Joe Buhler and Stan Wagon 'Secrets of the Madelung Constant' page 50 fig 1. British Library Cataloguing in Publication Data Mathews, P.C. Vector calculus.- (Springer undergraduate mathematics series). i. Vector analysis 2. Calculus of tensors I.Title 515.6'3 ISBN 3540761802 Library of Congress Cataloging-in-Publication Data Matthews, P.C. (Paul Charles), 1962- Vector calculus/ P.C. Matthews. cm- (Springer undergraduate mathematics series) Includes index. ISBN 3-540-76180-2 (pbk.: acid-free paper) 1. Vector analysis. I. Title. 11. Series 97-41191 QA433.M38 1998 515'.63-dc21 Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency.Enquiries concerning reproduction outside those terms should be sent to the publishers. Springer Undergraduate Mathematics Series ISSN 1615-2085 ISBN3-54o-7618o-2 Springer-Verlag London Berlin Heidelberg Springer Science+Business Media springeronline.com The use of registered names, trademarks etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or ommissions that may be made. Mathews: Vector Calculus ©1998, Springer-Verlag London Limited All rights reserved. No part of this publication maybe reproduced, stored in any electronic or mechanical form, including photocopy, recording or otherwise, without the prior written permission of the publisher. First Indian Reprint 2005 Second Indian Reprint 2008 ISBN 978-81-8128-295-8 This edition is manufactured in India for sale in India, Pakistan, Bangladesh, Nepal and Sri Lanka and any other country as authorized by the publisher. This edition is published by Springer (India) Private Limited, A part of Springer Science+Business Media, Registered Office: 906-907, Akash Deep Building, Barakhamba Road, New Delhi - no oo1, India. Printed in India by Rashtriya Printers, Delhi.

Preface Vector calculus is the fundamental language of mathematical physics. It pro- vides a way to describe physical quantities in three-dimensional space and the way in which these- quantities vary. Many topics in the physical sciences can be analysed mathematically using the techniques of vector calculus. These top- ics include fluid dynamics, solid mechanics and electromagnetism, all of which involve a description of vector and scalar quantities in three dimensions. This book assumes no previous knowledge of vectors. However, it is assumed that the reader has a knowledge of basic calculus, including differentiation, integration and partial differentiation. Some knowledge of linear algebra is also required, particularly the concepts of matrices and determinants. The book is designed to be self-contained, so that it is suitable for a pro- gramme of individual study. Each of the eight chapters introduces a new topic, and to facilitate understanding of the material, frequent reference is made to physical applications. The physical nature of the subject is clarified with over sixty diagrams, which provide an important aid to the comprehension of the new concepts. Following the introduction of each new topic, worked examples are provided. It is essential that these are studied carefully, so that a full un- derstanding is developed before moving ahead. Like much of mathematics, each section of the book is built on the foundations laid in the earlier sections and chapters. In addition to the worked examples, a section of exercises is included at the middle and at the end of each chapter. Solutions to all the exercises are given at the back of the book, but the student is encouraged to attempt all of the exercises before looking up the answers! At the end of each chapter, a one-page summary is given, listing the most essential points of the chapter.

Vi Vector Calculus The first chapter covers the basic concepts of vectors and scalars, the ways in which vectors can be multiplied together and some of the applications of vectors to physics and geometry. Chapter 2 defines the ways in which vector and scalar quantities can be integrated, covering line integrals, surface integrals and volume integrals. Again, these are illustrated with physical applications. Techniques for differentiating vectors and scalars are given in Chapter 3, which forms the essential core of the subject of vector calculus. The key concepts of gradient, divergence and curl are defined, which provide the basis for the following chapters. Chapter 4 introduces a new and powerful notation, suffix notation, for ma- nipulating complicated vector expressions. Quantities that run to several lines using conventional vector notation can be written extremely compactly using suffix notation. One of the main reasons for writing this book is that there are very few other books that make full use of suffix notation, although it is commonly used in undergraduate mathematics courses. Two important theorems, the divergence theorem and Stokes's theorem, are covered in Chapter 5. These help to tie the subject together, by providing links between the different forms of integrals from Chapter 2 and the derivatives of vectors from Chapter 3. Chapter 6 covers the general theory of orthogonal curvilinear coordinate systems and describes the two most important examples, cylindrical polar co- ordinates and spherical polar coordinates. Chapter 7 introduces a more rigorous, mathematical definition of vectors and scalars, which is based on the way in which they transform when the coordinate system is rotated. This definition is extended to a more general class of objects known as tensors. Some physical examples of tensors are given to aid the understanding of what can be a difficult concept to grasp. The final chapter gives a brief overview of some of the applications of the subject, including the flow of heat within a body, the mechanics of solids and fluids and electromagnetism.

Table of Contents 1 1 1 2 3 4 7 9 1. Vector Algebra ............................................. 1.1 Vectors and scalars ....................................... 1.1.1 Definition of a vector and a scalar .................... ................................ 1.1.2 Addition of vectors 1.1.3 Components of a vector ............................. 1.2 Dot product ............................................. 1.2.1 Applications of the dot product ...................... 1.3 Cross product ............................................ 1.3.1 Applications of the cross product ..................... 11 1.4 Scalar triple product ...................................... 14 1.5 Vector triple product ..................................... 16 1.6 Scalar fields and vector fields ............................... 17 2. Line, Surface and Volume Integrals ......................... 21 2.1 Applications and methods of integration ..................... 21 2.1.1 Examples of the use of integration .................... 21 Integration by substitution .......................... 22 2.1.2 Integration by parts ................................ 23 2.1.3 2.2 Line integrals ............................................ 25 Introductory example: work done against a force ....... 25 2.2.1 2.2.2 Evaluation of line integrals .......................... 26 2.2.3 Conservative vector fields ............................ 28 2.2.4 Other forms of line integrals ......................... 30 2.3 Surface integrals .......................................... 31 Introductory example: flow through a pipe ............. 31 2.3.1 2.3.2 Evaluation of surface integrals ........................ 33 2.3.3 Other forms of surface integrals ...................... 38 .......................................... 39 2.4 Volume integrals

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