Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis

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

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

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Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis