Tensors (eBook)

Anadi Das is a Professor Emeritus at Simon Fraser University, British Columbia, Canada.  He earned his Ph.D. in Mathematics and Physics from the National University of Ireland and his D.Sc. from Calcutta University.  He has published numerous papers in publications such as the Journal of Mathematical Physics and Foundation of Physics.  His book entitled The Special Theory of Relativity: A Mathematical Exposition was published by Springer in 1993.

Preface 6
Finite-Dimensional Vector Spaces and Linear Mappings 12
Fields 12
Finite-Dimensional Vector Spaces 14
Linear Mappings of a Vector Space 20
Dual or Covariant Vector Spaces 22
Tensor Algebra 27
Second-Order Tensors 27
Higher-Order Tensors 35
Exterior or Grassmann Algebra 42
Inner Product Vector Spacesand the Metric Tensor 53
Tensor Analysis on a Differentiable Manifold 63
Differentiable Manifolds 63
Tangent Vectors, Cotangent Vectors, and Parametrized Curves 71
Tensor Fields over Differentiable Manifolds 80
Differential Forms and Exterior Derivatives 91
Differentiable Manifolds with Connections 103
The Affine Connection and Covariant Derivative 103
Covariant Derivatives of Tensors along a Curve 112
Lie Bracket, Torsion, and Curvature Tensor 118
Riemannian and Pseudo-Riemannian Manifolds 132
Metric Tensor, Christoffel Symbols, and Ricci Rotation Coefficients 132
Covariant Derivatives and the Curvature Tensor 146
Curves, Frenet-Serret Formulas,and Geodesics 168
Special Coordinate Charts 192
Special Riemannian and Pseudo-Riemannian Manifolds 211
Flat Manifolds 211
The Space of Constant Curvature 216
Einstein Spaces 225
Conformally Flat Spaces 227
Hypersurfaces, Submanifolds, and Extrinsic Curvature 236
Two-Dimensional Surfaces Embedded in a Three-Dimensional Space 236
(N-1)-Dimensional Hypersurfaces 244
D-Dimensional Submanifolds 256
Appendix I Fibre Bundles 267
Appendix II Lie Derivatives 268
Answers and Hints to Selected Exercises 281
References 285
List of Symbols 288
Index 291