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

Written by: Ray M. Bowen (Texas A&M University) and C.C. Wang (Rice University)

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.”

### 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 