Introduction to Vectors and Tensors Second Edition--Two Volumes Bound as One

by Bowen, Ray M.; Wang, C.-C.

• ISBN13: 9780486469140
• ISBN10: 048646914X
• Edition: 2nd
• Format: Paperback
• Copyright: 2009-01-15
• Publisher: Dover Publications
• More Book Details

Summary

This convenient single-volume compilation of two texts offers both an introduction and an in-depth survey. Geared toward engineering and science students rather than mathematicians, its less rigorous treatment focuses on physics and engineering applications. A practical reference for professionals, it is suitable for advanced undergraduate and graduate students. 1976 edition.

Author Biography

Ray M. Bowen earned his Ph.D. in mechanical engineering from Texas A&M University and later served as president there. He is currently president emeritus with a faculty appointment in mechanical engineering.

Table of Contents

Linear and Multilinear Algebra
Contents of Volume 2	p. vii
Basic Mathematics
Selected Readings for Part I	p. 2
Elementary Matrix Theory	p. 3
Sets, Relations, and Functions	p. 13
Sets and Set Algebra	p. 13
Ordered Pairs" Cartesian Products" and Relations	p. 16
Functions	p. 18
Groups, Rings and Fields	p. 23
The Axioms for a Group	p. 23
Properties of a Group	p. 26
Group Homomorphisms	p. 29
Rings and Fields	p. 33
Vector and Tensor Algebra
Selected Readings for Part II	p. 40
Vector Spaces	p. 41
The Axioms for a Vector Space	p. 41
Linear Independence, Dimension and Basis	p. 46
Intersection, Sum and Direct Sum of Subspaces	p. 55
Factor Spaces	p. 60
Inner Product Spaces	p. 63
Orthogonal Bases and Orthogonal Compliments	p. 70
Reciprocal Basis and Change of Basis	p. 76
Linear Transformations	p. 85
Definition of a Linear Transformation	p. 85
Sums and Products of Linear Transformations	p. 93
Special Types of Linear Transformations	p. 97
The Adjoint of a Linear Transformation	p. 105
Component Formulas	p. 118
Determinants and Matrices	p. 125
The Generalized Kronecker Deltas and the Summation Convention	p. 125
Determinants	p. 130
The Matrix of a Linear Transformation	p. 136
Solution of Systems of Linear Equations	p. 142
Spectral Decompositions	p. 145
Direct Sum of Endomorphisms	p. 145
Eigenvectors and Eigenvalues	p. 148
The Characteristic Polynomial	p. 151
Spectral Decomposition for Hermitian Endomorphisms	p. 158
Illustrative Examples	p. 171
The Minimal Polynomial	p. 176
Spectral Decomposition for Arbitrary Endomorphisms	p. 182
Tensor Algebra	p. 203
Linear Functions, the Dual Space	p. 203
The Second Dual Space, Canonical Isomorphisms	p. 213
Multilinear Functions, Tensors	p. 218
Contractions	p. 229
Tensors on Inner Product Spaces	p. 235
Exterior Algebra	p. 247
Skew-Symmetric Tensors and Symmetric Tensors	p. 247
The Skew-Symmetric Operator	p. 250
The Wedge Product	p. 256
Product Bases and Strict Components	p. 263
Determinants and Orientations	p. 271
Duality	p. 280
Transformation to Contravariant Representation	p. 287
Index	p. ix
Vector and Tensor Analysis
Vector and Tensor Analysis
Selected Readings for Part III	p. 296
Euclidean Manifolds	p. 297
Euclidean Point Spaces	p. 297
Coordinate Systems	p. 306
Transformation Rules for Vector and Tensor Fields	p. 324
Anholonomic and Physical Components of Tensors	p. 332
Christoffel Symbols and Covariant Differentiation	p. 339
Covariant Derivatives along Curves	p. 353
Vector Fields and Differential Forms	p. 359
Lie Derivatives	p. 359
Frobenius Theorem	p. 368
Differential Forms and Exterior Derivative	p. 373
The Dual Form of Frobenius Theorem: the Poincare Lemma	p. 381
Vector Fields in a Three-Dimensional Euclidean Manifold, I. Invariants and Intrinsic Equations	p. 389
Vector Fields in a Three-Dimensional Euclidean Manifold, II. Representations for Special Class of Vector Fields	p. 399
Hypersurfaces in a Euclidean Manifold
Normal Vector, Tangent Plane, and Surface Metric	p. 407
Surface Covariant Derivatives	p. 416
Surface Geodesics and the Exponential Map	p. 425
Surface Curvature, I. The Formulas of Weingarten and Gauss	p. 433
Surface Curvature, II. The Riemann-Christoffel Tensor and the Ricci Identities	p. 443
Surface Curvature, III. The Equations of Gauss and Codazzi	p. 449
Surface Area, Minimal Surface	p. 454
Surfaces in a Three-Dimensional Euclidean Manifold	p. 457
Elements of Classical Continuous Groups
The General Linear Group and Its Subgroups	p. 463
The Parallelism of Cartan	p. 469
One-Parameter Groups and the Exponential Map	p. 476
Subgroups and Subalgebras	p. 482
Maximal Abelian Subgroups and Subalgebras	p. 486
Integration of Fields on Euclidean Manifolds, Hypersurfaces, and Continuous Groups
Are Length, Surface Area, and Volume	p. 491
Integration of Vector Fields and Tensor Fields	p. 499
Integration of Differential Forms	p. 503
Generalized Stokes' Theorem	p. 507
Invariant Integrals on Continuous Groups	p. 515
Index	p. vii
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