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 onesemester 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 ThreeDimensiona1 Euclidean Manifold, I. Invariants and Intrinsic Equations
 Section 54. Vector Fields in a ThreeDimensiona1 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 RiemannChristoffel 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 ThreeDimensional 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. OneParameter 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
Visit: Introduction to Vectors and Tensors, Vol 2: Vector and Tensor Analysis