INTRODUCTION TO VECTORS AND TENSORS Linear and Multilinear Algebra Volume 1 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 0-306-37508-7 (v. 1)) Updated 2010 ____________________________________________________________________________ PREFACE To Volume 1 This work represents our effort to present the basic concepts of vector and tensor analysis. Volume I begins with a brief discussion of algebraic structures followed by a rather detailed discussion of the algebra of vectors and tensors. Volume II begins with a discussion of Euclidean Manifolds which leads to a development of the analytical and geometrical aspects of vector and tensor fields. We have not included a discussion of general differentiable manifolds. However, we have included a chapter on vector and tensor fields defined on Hypersurfaces in a Euclidean Manifold. In preparing this two volume work our intention is to present to Engineering and Science students a modern introduction to vectors and tensors. Traditional courses on applied mathematics have emphasized problem solving techniques rather than the systematic development of concepts. As a result, it is possible for such courses to become terminal mathematics courses rather than courses which equip the student to develop his or her understanding further. As Engineering students our courses on vectors and tensors were taught in the traditional way. We learned to identify vectors and tensors by formal transformation rules rather than by their common mathematical structure. The subject seemed to consist of nothing but a collection of mathematical manipulations of long equations decorated by a multitude of subscripts and superscripts. Prior to our applying vector and tensor analysis to our research area of modern continuum mechanics, we almost had to relearn the subject. Therefore, one of our objectives in writing this book is to make available a modern introductory textbook suitable for the first in-depth exposure to vectors and tensors. Because of our interest in applications, it is our hope that this book will aid students in their efforts to use vectors and tensors in applied areas. The presentation of the basic mathematical concepts is, we hope, as clear and brief as possible without being overly abstract. Since we have written an introductory text, no attempt has been made to include every possible topic. The topics we have included tend to reflect our personal bias. We make no claim that there are not other introductory topics which could have been included. Basically the text was designed in order that each volume could be used in a one-semester course. We feel Volume I is suitable for an introductory linear algebra course of one semester. Given this course, or an equivalent, Volume II is suitable for a one semester course on vector and tensor analysis. Many exercises are included in each volume. However, it is likely that teachers will wish to generate additional exercises. Several times during the preparation of this book we taught a one semester course to students with a very limited background in linear algebra and no background in tensor analysis. Typically these students were majoring in Engineering or one of the Physical Sciences. However, we occasionally had students from the Social Sciences. For this one semester course, we covered the material in Chapters 0, 3, 4, 5, 7 and 8 from Volume I and selected topics from Chapters 9, 10, and 11 from Volume 2. As to level, our classes have contained juniors, iii iv PREFACE seniors and graduate students. These students seemed to experience no unusual difficulty with the material. It is a pleasure to acknowledge our indebtedness to our students for their help and forbearance. Also, we wish to thank the U. S. National Science Foundation for its support during the preparation of this work. We especially wish to express our appreciation for the patience and understanding of our wives and children during the extended period this work was in preparation. Houston, Texas R.M.B. C.-C.W. ______________________________________________________________________________ CONTENTS Vol. 1 Linear and Multilinear Algebra Contents of Volume 2………………………………………………………. vii PART I 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 II 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 v CONTENTS OF VOLUME 1 vi Section 16. Sums and Products of Linear Transformations……………… 93 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 . __________________________________________________________________________ CONTENTS Vol. 2 Vector and Tensor Analysis 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 Section 61. Surface Area, Minimal Surface 454 vii CONTENTS OF VOLUME 11 viii 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