One of several engineering and advanced algebra textbooks written and generously shared by Ray M. Bowen of Texas A&M University. Bowen was joined by fellow educator C.C. Wang in the writing of this textbook. Introduction to Vectors and Tensors, Vol 1 is the first of two volumes which discuss the basic concepts of vector and tensor analysis. The second volume Introduction to Vectors and Tensors, Vol 2. Vector and Tensor Analysis is also featured here on The Free Online Textbook List.
The goal of this textbook is to provide Engineering, Physical Science and Social Science students with “a modern introduction to vectors and tensors” with a focus on their applications. Volume 1 includes information often found in an introductory linear algebra course. Volume II covers vector and tensor analysis.
Bowen, a member of the mechanical engineering faculty, wrote this text to focus on the application of the topics rather than offering “a collection of mathematical manipulations of long equations decorated by a multitude of subscripts and superscripts.”
Table of Contents Introduction to Vectors and Tensors, Vol.1
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PART I BASIC MATHEMATICS
- CHAPTER 0 Elementary Matrix Theory
- CHAPTER 1 Sets, Relations, and Functions
- Section 1. Sets and Set Algebra
- Section 2. Ordered Pairs” Cartesian Products” and Relations
- Section 3. Functions
- CHAPTER 2 Groups, Rings and Fields
- Section 4. The Axioms for a Group
- Section 5. Properties of a Group
- Section 6. Group Homomorphisms
- Section 7. Rings and Fields
- CHAPTER 3 Vector Spaces
- Section 8. The Axioms for a Vector Space
- Section 9. Linear Independence, Dimension and Basis
- Section 10. Intersection, Sum and Direct Sum of Subspaces
- Section 11. Factor Spaces
- Section 12. Inner Product Spaces
- Section 13. Orthogonal Bases and Orthogonal Compliments
- Section 14. Reciprocal Basis and Change of Basis
- CHAPTER 4. Linear Transformations
- Section 15. Definition of a Linear Transformation
- Section 16. Sums and Products of Linear Transformations
- Section 17. Special Types of Linear Transformations
- Section 18. The Adjoint of a Linear Transformation
- Section 19. Component Formulas
- CHAPTER 5. Determinants and Matrices
- Section 20.The Generalized Kronecker Deltas and the Summation Convention
- Section 21. Determinants
- Section 22. The Matrix of a Linear Transformation
- Section 23. Solution of Systems of Linear Equations
- CHAPTER 6 Spectral Decompositions
- Section 24. Direct Sum of Endomorphisms
- Section 25. Eigenvectors and Eigenvalues
- Section 26. The Characteristic Polynomial
- Section 27. Spectral Decomposition for Hermitian Endomorphisms
- Section 28. Illustrative Examples
- Section 29. The Minimal Polynomial
- Section 30. Spectral Decomposition for Arbitrary Endomorphisms
- CHAPTER 7 Tensor Algebra
- Section 31. Linear Functions, the Dual Space
- Section 32. The Second Dual Space, Canonical Isomorphisms
- Section 33. Multilinear Functions, Tensors
- Section 34. Contractions
- Section 35. Tensors on Inner Product Spaces
- CHAPTER 8. Exterior Algebra
- Section 36. Skew-Symmetric Tensors and Symmetric Tensors
- Section 37. The Skew-Symmetric Operator
- Section 38. The Wedge Product
- Section 39. Product Bases and Strict Components
- Section 40. Determinants and Orientations
- Section 41. Duality
- Section 42. Transformation to Contravariant Representation
PART II VECTOR AND TENSOR ALGEBRA
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Introduction to Vectors and Tensors, Vol 1 Linear and Multilinear Algebra