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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)) ____________________________________________________________________________ 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 __________________________________________________________________________ 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 ______________________________________________________________________________ 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 _____________________________________________________________________________ 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
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