Here is a modern introduction to the theory of tensor algebra and tensor analysis. It discusses tensor algebra and introduces differential manifold. Coverage also details tensor analysis, differential forms, connection forms, and curvature tensor. In addition, the book investigates Riemannian and pseudo-Riemannian manifolds in great detail. Throughout, examples and problems are furnished from the theory of relativity and continuum mechanics.
Author(s): Anadi Jiban Das
Edition: 1
Year: 2007
Language: English
Pages: 300
Contents......Page 8
Preface......Page 6
List of Figures......Page 10
1.1 Fields......Page 12
1.2 Finite-Dimensional Vector Spaces......Page 14
1.3 LinearMappings of a Vector Space......Page 20
1.4 Dual or Covariant Vector Spaces......Page 22
2.1 Second-Order Tensors......Page 27
2.2 Higher-Order Tensors......Page 35
2.3 Exterior or Grassmann Algebra......Page 42
2.4 Inner Product Vector Spaces and the Metric Tensor......Page 53
3.1 Differentiable Manifolds......Page 63
3.2 Tangent Vectors, Cotangent Vectors, and Parametrized Curves......Page 71
3.3 Tensor Fields over Differentiable Manifolds......Page 80
3.4 Differential Forms and Exterior Derivatives......Page 91
4.1 The Affine Connection and Covariant Derivative......Page 103
4.2 Covariant Derivatives of Tensors along a Curve......Page 112
4.3 Lie Bracket, Torsion, and Curvature Tensor......Page 118
5.1 Metric Tensor, Christoffel Symbols, and Ricci Rotation Coefficients......Page 132
5.2 Covariant Derivatives and the Curvature Tensor......Page 146
5.3 Curves, Frenet-Serret Formulas, and Geodesics......Page 168
5.4 Special Coordinate Charts......Page 192
6.1 Flat Manifolds......Page 211
6.2 The Space of Constant Curvature......Page 216
6.3 Einstein Spaces......Page 225
6.4 Conformally Flat Spaces......Page 227
7.1 Two-Dimensional Surfaces Embedded in a Three-Dimensional Space......Page 236
7.2 (N – 1)-Dimensional Hypersurfaces......Page 244
7.3 D-Dimensional Submanifolds......Page 256
Appendix 1. Fibre Bundles......Page 268
Appendix 2. Lie Derivatives......Page 274
Answers and Hints to Selected Exercises......Page 282
References......Page 288
List of Symbols......Page 291
C......Page 296
G......Page 297
N......Page 298
S......Page 299
Z......Page 300
