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SCHAUM'S OUTLINE OF

THEORY AND PROBLEMS OF

TENSOR CALCULUS •

DAVID C. KAY, Ph.D. Profe.ssor of A1a,hematics Universi,)' of North Carolino. Ctt Asheville

McGRAW-HILL f't'ew York Son Fra�C() Wo.,.hi"8ton, D.C. AUi;klon.d BogotO CoMc-CU Wndo,1 ,�fatlrid ,r.nro City Afilan &fon.trf!al 1Vew Df:hli Snn Juan Sm,tJtiport: Syd,u1y Tok yo Toronto

DAVID C. KAY is currently Professor and Chairman of Mathematics at the University of North Carolina at Asheville; formerly he taught in the graduate program at the University of Oklahoma for 17 years. He received his Ph.D. in geometry at Michigan State University in 1963. He is the author of more than 30 articles in the areas of distance geometry, convexity theory, and related functional analysis. Schaum's Outline of Theory and Problems of TENSOR CALCULUS Copyright© 1988 by The McGraw-Hill Companies, Inc. All rights reserved. Printed in the United States of America. Except as permitted under the Copyright Act of 1976, no part of this publication may be reproduced or distributed in any form or by any means, or stored in a data base or retrieval system, without the prior written permission of the publisher. 14 15 16 17 18 19 20 cus cus 09 08 07 06 05

ISBN 0-07-033484-6 Sponsoring Editor, David Beckwith Production Supervisor, Denise Puryear Editing Supervisor, Marthe Grice

Library of Congress Cataloging-in-Publication Data Kay, David C. Schaum's outline of theory and problems of tensor calculus. (Schaum's Outline series) 1. Calculus of tensors-Problems, exercises, etc. I. Title. II. Title: Theory and problems of tensor calculus. QA433.K39 1988 515'.63 87-32515 ISBN 0-07-033484-6

McGraw-Hill A Division ofTheMcGraw·HiUCompanies

Preface This Outline is designed for use by both undergraduates and graduates who find they need to master the basic methods and concepts of tensors. The material is written from both an elementary and applied point of view, in order to provide a lucid introduction to the subject. The material is of fundamental importance to theoretical physics (e.g., field and electromagnetic theory) and to certain areas of engineering (e.g., aerodynamics and fluid mechanics). W henever a change of coordinates emerges as a satisfactory way to solve a problem, the subject of tensors is an immediate requisite. Indeed, many techniques in partial differential equations are tensor transformations in disguise. W hile physicists readily recog­ nize the importance and utility of tensors, many mathematicians do not. It is hoped that the solved problems of this book will allow all readers to find out what tensors have to offer them. Since there are two avenues to tensors and since there is general disagreement over which is the better approach for beginners, any author has a major decision to make. After many hours in the classroom it is the author's opinion that the tensor component approach (replete with subscripts and superscripts) is the correct one to use for beginners, even though it may require some painful initial adjustments. Although the more sophisticated, noncomponent approach is neces­ sary for modern applications of the subject, it is believed that a student will appreciate and have an immensely deeper understanding of this sophisticated approach to tensors after a mastery of the component approach. In fact, noncomponent advocates frequently surrender to the introduction of components after all; some proofs and important tensor results just do not lend themselves to a completely component-free treatment. The Outline follows, then, the tradition­ al component approach, except in the closing Chapter 13, which sketches the more modern treatment. Material that extends Chapter 13 to a readable introduc­ tion to the geometry of manifolds may be obtained, at cost, by writing to the author at: University of North Carolina at Asheville, One University Heights, Asheville, NC 28804-3299. The author has been strongly influenced over the years by the following major sources of material on tensors and relativity: J. Gerretsen, Lectures on Tensor Calculus and Differential Geometry, P. Noordhoff: Goningen, 1962. I. S. Sokolnikoff, Tensor Analysis and Its Applications, McGraw-Hill: New York, 1950. Synge and Schild, Tensor Calculus, Toronto Press: Toronto, 1949. W. Pauli, Jr., Theory of Relativity, Pergamon: New York; 1958. R. D. Sard, Relativistic Mechanics, W. A. Benjamin: New York, 1970. Bishop and Goldberg, Tensor Analysis on Manifolds, Macmillan: New York, 1968. Of course, the definitive work from the geometrical point of view is L. P. Eisenhart, Riemannian Geometry, Princeton University Press: Princeton, N.J., 1949. The author would like to acknowledge significant help in ferreting out typographical errors and other imperfections by the readers: Ronald D. Sand-

PREFACE

strom, Professor of Mathematics at Fort Hays State University, and John K. Beem, Professor of Mathematics at the University of Missouri. Appreciation is also extended to the editor, David Beckwith, for many helpful suggestions. DAVID C. KAY

Contents Chapter

1 THE EINSTEIN SUMMATION CONVENTION. . . .. . . . . . . ..... . .. . . . . . Introduction . .. . .... . . . . . ... ..... .... . ... ... . . ..... . .... .. . . . . . . . Repeated Indices in Sums ...... . . ... .. .. . .. .. . .. ... .. . . . . . . . . .. . . . . Double Sums .. . ...... . . ...... .... . . .. . . .. . ... .. . .. . . . .... . ... . . . Substitutions ... .... .... .... . . .. . . .... .... .. .... . .... . . . .. . ... . . . . Kronecker Delta and Algebraic Manipulations. . . .... ... . . . . . ... . . .. . . . .

1 1 2 2 3

Chapter 2

BASIC LINEAR ALGEBRA FOR TENSORS . . . . . . . . . . . . . ... . . . . . .. . . .

8

Introduction .... . ... ....... ....... .. .. .. .... . . . .... . ..... . . . .. . . . Tensor Notation for Matrices, Vectors, and Determinants. .. . . . . . .. . . . . . . . Inverting a Matrix ..... . ...... . .... . . ... . . .. . .. . . . . ... . . . . . . . . . . .. Matrix Expressions for Linear Systems and Quadratic Forms . . . . ... . . . . . . . Linear Transformations ... ... . .. . . .. .. .... ..... . .... . . . . . .. . . . . . . . . General Coordinate Transformations. . . ... ... . . . . . .. . . . . . . .. . . . . . . . . . . The Chain Rule for Partial Derivatives..... . .... . .. . . .... . . . . . . . . . . . ..

8 8 10 10 11 12 13

Chapter 3

GENERAL TENSORS...............................................

23

3.1 3.2 3.3 3.4 3.5 3.6

Coordinate Transformations.. ...... . . . . .... . ... . . . .. ... .. .... . . . . . .. First-Order Tensors . .. . .... . . .... . ... .. .. . ... ... . . .. ... .. . . . . . . .. . Invariants . . ...... . . .. . ... . . .. ... . . . .... . . .. . .... .. . . . . . . . . . . . . .. Higher-Order Tensors .. . .. . . . . . . . ... ... ... ... . . . . . . . . . . . . . . . . . . . . . The Stress Tensor. ... . . . . . . . . .. . . .. .. . . .. . . . . . . . . . . . . . . . . .. . . . . . .. Cartesian Tensors . .. .. .. . . ..... ... . . . ... . .... . ... .. .. . . . . . . . . . . . ..

23 26 28 29 29 31

4 TENSOR OPERATIONS: TESTS FOR TENSOR CHARACTER . . ... . . . .

43

4.1 Fundamental Operations .. . .... . . . . . . . .... . . . ... .. ... . . .. . . . . . . . . . . 4.2 Tests for Tensor Character . .. .. . . .. . . ... . . .... . .. . ... . . . . . . . . . . .. . . 4.3 Tensor Equations . . . .......... . . .. .. . .... ... . . . . . . . .. . . . . . . . . . . . ..

43 45 45

Chapter 5 THE METRIC TENSOR . . . . . . . . . .... . . . . . .. . .. . . .. . ... . . . . .... . . . . .

51

Chapter

1.1 1.2 1.3 1.4 1.5

1

2.1 2.2 2.3 2.4 2.5 2.6 2.7

5.1 5 .2 5.3 5.4 5.5 5.6

Introduction . . . . . . . .. . . ... . . . . . .. . . . . . . . . . . . .. . ... . . . .. . . . . . . . . . . Arc Length in Euclidean Space ............... , . .. . .... . . . ... . . .. . . .. Generalized Metrics; The Metric Tensor. . . . . . .. . . . . . . . . .. . . . . . . . . . . . . . Conjugate Metric Tensor; Raising and Lowering Indices. ...... . . . . . . . . . . . Generalized Inner-Product Spaces. . .. . .. . .. . ..... ..... . . . . . . . . . . . . . . . Concepts of Length and Angle ... . . ...... .. ....... . .... . . . . .. . . . . .. .

51 51 52 55 55 56

CONTENTS

Chapter

6 THE DERIVATIV E OF A TENSOR .................................. 6.1 6.2 6.3 6.4 6.5 6.6

Inadequacy of Ordinary Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christoffel Symbols of the First Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Christoffel Symbols of the Second Kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Covariant Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Absolute Differentiation along a Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rules for Tensor Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68 68 68 70 71 72 74

Chapter

7

RIEMANNIAN GEOMETRY OF CURV ES............................ 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Length and Angle under an Indefinite Metric . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Null Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Regular Curves: Unit Tangent Vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Regular Curves: Unit Principal Normal and Curvature . . . . . . . . . . . . . . . . . . . 7.6 Geodesics as Shortest Arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 83 84 85 86 88

Chapter

8

RIEMANNIAN CURVATURE........................................ 8.1 The Riemann Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Properties of the Riemann Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Riemannian Curvature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 The Ricci Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101 101 101 103 105

Chapter

9 SPACES OF CONSTANT CURVATURE; NORMAL COORDINATES .... 114 9.1 9.2 9.3 9.4 9.5

Chapter

10

Zero Curvature and the Euclidean Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Flat Riemannian Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Schur's Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Einstein Tensor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

114 116 117 119 119

TENSORS IN EUCLIDEAN GEOMETRY............................. 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Curve Theory; The Moving Frame . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Curvature and Torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Regular Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Parametric Lines; Tangent Space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6 First Fundamental Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7 Geodesics on a Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8 Second Fundamental Form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9 Structure Formulas for Surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 127 127 130 130 132 133 135 136 137 138

CONTENTS

Chapter

11

TENSORS IN CLASSICAL MECHANICS........... . ................. 154 11.1 11.2 11.3 11.4 11.5

Chapter

12

13

154 154 155 156 157

TENSORS IN SPECIAL RELATIVITY ............. . ................. 164 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8

Chapter

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Particle Kinematics in Rectangular Coordinates . . . . . . . . . . . . . . . . . . . . . . . . Particle Kinematics in Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . Newton's Second Law in Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . Divergence, Laplacian, Curl. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Event Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Lorentz Group and the Metric of SR . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simple Lorentz Matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Physical Implications of the Simple Lorentz Transformation . . . . . . . . . . . . . . Relativistic Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic Mass, Force, and Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Maxwell's Equations in SR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

164 164 166 167 169 169 171 172

TENSOR FIELDS ON MANIFOLDS.................................. 189 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract Vector Spaces and the Group Concept. . . . . . . . . . . . . . . . . . . . . . . . Important Concepts for Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Algebraic Dual of a Vector Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tensors on Vector Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tangent Space; Vector Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

189 189 190 191 193 194 197 199

ANSWERS TO SUPPLEMENTARY PROBLEMS ................................... 213 INDEX ......................................................................... 223

Chapter 1 The Einstein Summation Convention 1.1 INTRODUCTION A study of tensor calculus requires a certain amount of background material that may seem unimportant in itself, but without which one could not proceed very far. Included in that prerequisite material is the topic of the present chapter, the summation convention. As the reader proceeds to later chapters he or she will see that it is this convention which makes the results of tensor analysis surveyable. 1.2 REPEATED INDICES IN SUMS A certain notation introduced by Einstein in his development of the Theory of Relativity streamlines many common algebraic expressions. Instead of using the traditional sigma for sums, the strategy is to allow the repeated subscript to become itself the designation for the summation. Thus, alxl + azXz + a3X3 + ... + anxn =

I

a;X; i=l becomes just a;x;, where 1 � i � n is adopted as the universal range for summation. EXAMPLE 1.1 The expression a;Jxk does not indicate summation, but both respective ranges 1 ;§ i ;§ n and 1 ;§ j ;§ n. If n = 4, then

a;;X k

and

a;1x1

do so over the

= a11xk + a22 k + a33xk + a44xk a;jxJ = a; x + a;2x2 + a;3x3 + a; 4 4

aiixk

X

1

X

1

Free and Dummy Indices In Example 1.1, the expression a ij xj involves two sorts of indices. The index of summation, j, which ranges over the integers 1, 2, 3, . . . , n, cannot be preempted. But at the same time, it is clear that the use of the particular character j is inessential; e.g., the expressions a;,x, and a;vx v represent exactly the same sum as a;jxj does. For this reason, j is called a dummy index. The index i, which may take on any particular value 1, 2, 3, ... , n independently, is called a free index. Note that, although we call the index i "free" in the expression a;jxj , that "freedom" is limited in the sense that generally, unless i = k,

EXAMPLE 1.2 If n = 3, write down explicitly the equations represented by the expression Y; =

a; ,x,.

Holding i fixed and summing over r = 1, 2, 3 yields Next, setting the free index i = 1, 2, 3 leads to three separate equations: Yi

=

auxi

Y2

=

YJ

= a31X1

a21X1

+ a12X2 + a13 3 + a22 2 + a23 3 + a32X2 + a33 X3 X

X

X

Einstein Summation Convention Any expression involving a twice-repeated index (occurring twice as a subscript, twice as a superscript, or once as a subscript and once as a superscript) shall automatically stand for its sum

1

2

THE EINSTEIN SUMMATION CONVENTION

[CHAP. 1

over the values 1, 2, 3, ... , n of the repeated index. Unless explicitly stated otherwise, the single exception to this rule is the character n, which represents the range of all summations. Remark 1:

Any free index in an expression shall have the same range as summation indices, unless stated otherwise.

Remark 2: No index may occur more than twice in any given expression. EXAMPLE 1.3 (a) According to Remark 2, an expression like a;;X; is without meaning. (b) The meaningless expression a�x;X; might be presumed to represent a�(x;)2, which is meaningful. (c) An expression of the form a;(x; + y;) is considered well-defined, for it is obtained by composition of the meaningful expressions a;z; and X; + Y; = z;. In other words, the index i is regarded as occuring once in the term (x; + y;).

1.3 DOUBLE SUMS An expression can involve more than one summation index. For example, a;jx i yj indicates a summation taking place on both i and j simultaneously. If an expression has two summation 2 ) indices, there will be a total of n terms in the sum; if there are three indices, there will be (dummy 3 n terms; and so on. The expansion of a;jx i yj can be arrived at logically by first summing over i, then over j: aijx iyj = aljxly j + a2jx2yj + a jx yj + · · · + anjXn Yj = (a11X1 Y1 + a12X1 Y2 + ... + a1nX1 Yn ) + (a21X2 Y1 + a22X2 Y2 + ... + a2nX2 Yn ) + (a 3 1X 3 Y1 + a 2X Y2 + ... + a nX Yn ) 3

3

3

3

3

[summed over i] [summed over j]

3

The result is the same if one sums over j first, and then over i. EXAMPLE 1.4 If n = 2, the expression Y; = c;a,s xs stands for the two equations:

y1 = c�a11 x1 + c�a2 1 x1 + c�a12 x2 + c� a22 x2 y2 = c�a1 1 x1 + c�a2 1 x1 + c�a12x2 + c� a22 x2

1.4 SUBSTITUTIONS Suppose it is required to substitute Y; = a ijxj in the equation Q = bijy ixj. Disregard of Remark 2 above would lead to an absurd expression like Q = b;ja ijxjxj. The correct procedure is first to identify any dummy indices in the expression to be substituted that coincide with indices occurring in the main expression. Changing these dummy indices to characters not found in the main expression, one may then carry out the substitution in the usual fashion. STEP STEP STEP

1 2 3

Q = b;jy ixj , Y; = a;h [dummy index j is duplicated] Y; = a;,x, [change dummy index from j to r] [substitute and rearrange] Q = b;/a;rx,)xj = a;,b;jx rxj

EXAMPLE 1.5 If Y; = a;i xi, express the quadratic form Q = g;i Y;Yi in terms of the x-variables.

First write: Y; = a;,x,, yi = ais xs . Then, by substitution,