Introduction to Vectors and Tensors Volume 2: Vector and Vector Analysis

246 Pages·1.135 MB·English

by  Bowen, Ray M., Bowen, Ray M., Wang, Chao-Cheng

INTRODUCTION TO VECTORS AND TENSORS

Vector and Tensor Analysis

Volume 2

Ray M. Bowen
Mechanical Engineering
Texas A&M University
College Station, Texas

and

C.-C. Wang
Mathematical Sciences
Rice University
Houston, Texas

Copyright Ray M. Bowen and C.-C. Wang
(ISBN 0-306-37509-5 (v. 2))

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PREFACE

To Volume 2

This is the second volume of a two-volume work on vectors and tensors. Volume 1 is concerned
with the algebra of vectors and tensors, while this volume is concerned with the geometrical
aspects of vectors and tensors. This volume begins with a discussion of Euclidean manifolds. The
principal mathematical entity considered in this volume is a field, which is defined on a domain in a
Euclidean manifold. The values of the field may be vectors or tensors. We investigate results due
to the distribution of the vector or tensor values of the field on its domain. While we do not discuss
general differentiable manifolds, we do include a chapter on vector and tensor fields defined on
hypersurfaces in a Euclidean manifold.

This volume contains frequent references to Volume 1. However, references are limited to
basic algebraic concepts, and a student with a modest background in linear algebra should be able
to utilize this volume as an independent textbook. As indicated in the preface to Volume 1, this
volume is suitable for a one-semester course on vector and tensor analysis. On occasions when we
have taught a one –semester course, we covered material from Chapters 9, 10, and 11 of this
volume. This course also covered the material in Chapters 0,3,4,5, and 8 from Volume 1.

We wish to thank the U.S. National Science Foundation for its support during the
preparation of this work. We also wish to take this opportunity to thank Dr. Kurt Reinicke for
critically checking the entire manuscript and offering improvements on many points.

Houston, Texas R.M.B.
C.-C.W.

iii

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CONTENTS

Vol. 2 Vector and Tensor Analysis

Contents of Volume 1………………………………………………… vii

PART III. VECTOR AND TENSOR ANALYSIS

Selected Readings for Part III………………………………………… 296

CHAPTER 9. Euclidean Manifolds………………………………………….. 297

Section 43. Euclidean Point Spaces……………………………….. 297
Section 44. Coordinate Systems…………………………………… 306
Section 45. Transformation Rules for Vector and Tensor Fields…. 324
Section 46. Anholonomic and Physical Components of Tensors…. 332
Section 47. Christoffel Symbols and Covariant Differentiation…... 339
Section 48. Covariant Derivatives along Curves………………….. 353

CHAPTER 10. Vector Fields and Differential Forms………………………... 359

Section 49. Lie Derivatives……………………………………….. 359
Section 5O. Frobenius Theorem…………………………………… 368
Section 51. Differential Forms and Exterior Derivative………….. 373
Section 52. The Dual Form of Frobenius Theorem: the Poincaré
Lemma……………………………………………….. 381
Section 53. Vector Fields in a Three-Dimensiona1 Euclidean
Manifold, I. Invariants and Intrinsic Equations…….. 389
Section 54. Vector Fields in a Three-Dimensiona1 Euclidean
Manifold, II. Representations for Special
Class of Vector Fields………………………………. 399

CHAPTER 11. Hypersurfaces in a Euclidean Manifold

Section 55. Normal Vector, Tangent Plane, and Surface Metric… 407
Section 56. Surface Covariant Derivatives………………………. 416
Section 57. Surface Geodesics and the Exponential Map……….. 425
Section 58. Surface Curvature, I. The Formulas of Weingarten
and Gauss…………………………………………… 433
Section 59. Surface Curvature, II. The Riemann-Christoffel
Tensor and the Ricci Identities……………………... 443
Section 60. Surface Curvature, III. The Equations of Gauss and Codazzi 449
v
vi CONTENTS OF VOLUME 2
Section 61. Surface Area, Minimal Surface………........................ 454
Section 62. Surfaces in a Three-Dimensional Euclidean Manifold. 457

CHAPTER 12. Elements of Classical Continuous Groups

Section 63. The General Linear Group and Its Subgroups……….. 463
Section 64. The Parallelism of Cartan……………………………. 469
Section 65. One-Parameter Groups and the Exponential Map…… 476
Section 66. Subgroups and Subalgebras…………………………. 482
Section 67. Maximal Abelian Subgroups and Subalgebras……… 486

CHAPTER 13. Integration of Fields on Euclidean Manifolds, Hypersurfaces, and
Continuous Groups

Section 68. Arc Length, Surface Area, and Volume……………... 491
Section 69. Integration of Vector Fields and Tensor Fields……… 499
Section 70. Integration of Differential Forms……………………. 503
Section 71. Generalized Stokes’ Theorem……………………….. 507
Section 72. Invariant Integrals on Continuous Groups…………... 515

INDEX………………………………………………………………………. x
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CONTENTS

Vol. 1 Linear and Multilinear Algebra

PART 1 BASIC MATHEMATICS

Selected Readings for Part I………………………………………………………… 2

CHAPTER 0 Elementary Matrix Theory…………………………………………. 3

CHAPTER 1 Sets, Relations, and Functions……………………………………… 13

Section 1. Sets and Set Algebra………………………………………... 13
Section 2. Ordered Pairs" Cartesian Products" and Relations…………. 16
Section 3. Functions……………………………………………………. 18

CHAPTER 2 Groups, Rings and Fields…………………………………………… 23

Section 4. The Axioms for a Group……………………………………. 23
Section 5. Properties of a Group……………………………………….. 26
Section 6. Group Homomorphisms…………………………………….. 29
Section 7. Rings and Fields…………………………………………….. 33

PART I1 VECTOR AND TENSOR ALGEBRA

Selected Readings for Part II………………………………………………………… 40

CHAPTER 3 Vector Spaces……………………………………………………….. 41

Section 8. The Axioms for a Vector Space…………………………….. 41
Section 9. Linear Independence, Dimension and Basis…………….….. 46
Section 10. Intersection, Sum and Direct Sum of Subspaces……………. 55
Section 11. Factor Spaces………………………………………………... 59
Section 12. Inner Product Spaces………………………..………………. 62
Section 13. Orthogonal Bases and Orthogonal Compliments…………… 69
Section 14. Reciprocal Basis and Change of Basis……………………… 75

CHAPTER 4. Linear Transformations……………………………………………… 85

Section 15. Definition of a Linear Transformation………………………. 85
Section 16. Sums and Products of Linear Transformations……………… 93
viii CONTENTS OF VOLUME 2
Section 17. Special Types of Linear Transformations…………………… 97
Section 18. The Adjoint of a Linear Transformation…………………….. 105
Section 19. Component Formulas………………………………………... 118

CHAPTER 5. Determinants and Matrices…………………………………………… 125

Section 20. The Generalized Kronecker Deltas
and the Summation Convention……………………………… 125
Section 21. Determinants…………………………………………………. 130
Section 22. The Matrix of a Linear Transformation……………………… 136
Section 23 Solution of Systems of Linear Equations…………………….. 142

CHAPTER 6 Spectral Decompositions……………………………………………... 145

Section 24. Direct Sum of Endomorphisms……………………………… 145
Section 25. Eigenvectors and Eigenvalues……………………………….. 148
Section 26. The Characteristic Polynomial………………………………. 151
Section 27. Spectral Decomposition for Hermitian Endomorphisms…….. 158
Section 28. Illustrative Examples…………………………………………. 171
Section 29. The Minimal Polynomial……………………………..……… 176
Section 30. Spectral Decomposition for Arbitrary Endomorphisms….….. 182

CHAPTER 7. Tensor Algebra………………………………………………………. 203

Section 31. Linear Functions, the Dual Space…………………………… 203
Section 32. The Second Dual Space, Canonical Isomorphisms…………. 213
Section 33. Multilinear Functions, Tensors…………………………..….. 218
Section 34. Contractions…......................................................................... 229
Section 35. Tensors on Inner Product Spaces……………………………. 235

CHAPTER 8. Exterior Algebra……………………………………………………... 247

Section 36. Skew-Symmetric Tensors and Symmetric Tensors………….. 247
Section 37. The Skew-Symmetric Operator……………………………… 250
Section 38. The Wedge Product………………………………………….. 256
Section 39. Product Bases and Strict Components……………………….. 263
Section 40. Determinants and Orientations………………………………. 271
Section 41. Duality……………………………………………………….. 280
Section 42. Transformation to Contravariant Representation……………. 287

INDEX…………………………………………………………………………………….x
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PART III

VECTOR AND TENSOR ANALYSIS
Selected Reading for Part III

BISHOP, R. L., and R. J. CRITTENDEN, Geometry of Manifolds, Academic Press, New York, 1964
BISHOP, R. L., and S. I. GOLDBERG, Tensor Analysis on Manifolds, Macmillan, New York, 1968.
CHEVALLEY, C., Theory of Lie Groups, Princeton University Press, Princeton, New Jersey, 1946
COHN, P. M., Lie Groups, Cambridge University Press, Cambridge, 1965.
EISENHART, L. P., Riemannian Geometry, Princeton University Press, Princeton, New Jersey, 1925.
ERICKSEN, J. L., Tensor Fields, an appendix in the Classical Field Theories, Vol. III/1.
Encyclopedia of Physics, Springer-Verlag, Berlin-Gottingen-Heidelberg, 1960.
FLANDERS, H., Differential Forms with Applications in the Physical Sciences, Academic Press,
New York, 1963.
KOBAYASHI, S., and K. NOMIZU, Foundations of Differential Geometry, Vols. I and II, Interscience,
New York, 1963, 1969.
LOOMIS, L. H., and S. STERNBERG, Advanced Calculus, Addison-Wesley, Reading, Massachusetts,
1968.
MCCONNEL, A. J., Applications of Tensor Analysis, Dover Publications, New York, 1957.
NELSON, E., Tensor Analysis, Princeton University Press, Princeton, New Jersey, 1967.
NICKERSON, H. K., D. C. SPENCER, and N. E. STEENROD, Advanced Calculus, D. Van Nostrand,
Princeton, New Jersey, 1958.
SCHOUTEN, J. A., Ricci Calculus, 2nd ed., Springer-Verlag, Berlin, 1954.
STERNBERG, S., Lectures on Differential Geometry, Prentice-Hall, Englewood Cliffs, New Jersey,
1964.
WEATHERBURN, C. E., An Introduction to Riemannian Geometry and the Tensor Calculus,
Cambridge University Press, Cambridge, 1957.