This first volume of this free online textbook for engineering and science students is covered here – Introduction to Vectors and Tensors, Vol 1: Linear and Multilinear Algebra. Both volumes are the work of Ray M. Bowen and C.C. Wang.
This text does refer to the first volume, but the author assures us that students who possess a “modest” background in linear algebra should be able to use this textbook. [Just in case you might need a refresher, you can refer to Volume 1 or check out the other Free Online Linear Algebra Textbooks featured on The Free Textbook List.]
This textbook is suitable for a one-semester course in vector and tensor analysis. Much of this textbook discusses Euclidean manifolds and the principle mathematical entity of fields. The authors caution that the text does not discuss “general differentiable manifolds” but “do include a chapter on vector and tensor fields defined on hypersurfaces in a Euclidean manifold.”
Table of Contents for Introduction to Vectors and Tensors, Vol 2
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PART III. VECTOR AND TENSOR ANALYSIS
- CHAPTER 9. Euclidean Manifolds
- Section 43. Euclidean Point Spaces
- Section 44. Coordinate Systems
- Section 45. Transformation Rules for Vector and Tensor Fields
- Section 46. Anholonomic and Physical Components of Tensors
- Section 47. Christoffel Symbols and Covariant Differentiation
- Section 48. Covariant Derivatives along Curves
- CHAPTER 10. Vector Fields and Differential Forms
- Section 49. Lie Derivatives
- Section 5O. Frobenius Theorem
- Section 51. Differential Forms and Exterior Derivative
- Section 52. The Dual Form of Frobenius Theorem: the Poincaré Lemma
- Section 53. Vector Fields in a Three-Dimensiona1 Euclidean Manifold, I. Invariants and Intrinsic Equations
- Section 54. Vector Fields in a Three-Dimensiona1 Euclidean Manifold, II. Representations for Special Class of Vector Fields
- CHAPTER 11. Hypersurfaces in a Euclidean Manifold
- Section 55. Normal Vector, Tangent Plane, and Surface Metric
- Section 56. Surface Covariant Derivatives
- Section 57. Surface Geodesics and the Exponential Map
- Section 58. Surface Curvature, I. The Formulas of Weingarten and Gauss
- Section 59. Surface Curvature, II. The Riemann-Christoffel Tensor and the Ricci Identities
- Section 60. Surface Curvature, III. The Equations of Gauss and Codazzi
- Section 61. Surface Area, Minimal Surface
- Section 62. Surfaces in a Three-Dimensional Euclidean Manifold
- CHAPTER 12. Elements of Classical Continuous Groups
- Section 63. The General Linear Group and Its Subgroups
- Section 64. The Parallelism of Cartan
- Section 65. One-Parameter Groups and the Exponential Map
- Section 66. Subgroups and Subalgebras
- Section 67. Maximal Abelian Subgroups and Subalgebras
- CHAPTER 13. Integration of Fields on Euclidean Manifolds, Hypersurfaces, and Continuous Groups
- Section 68. Arc Length, Surface Area, and Volume
- Section 69. Integration of Vector Fields and Tensor Fields
- Section 70. Integration of Differential Forms
- Section 71. Generalized Stokes’ Theorem
- Section 72. Invariant Integrals on Continuous Groups
- Index
View this Free Online Material at the source:
Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis