Tensors (eBook)

The Mathematics of Relativity Theory and Continuum Mechanics

Anadi Jiban Das (Autor)

eBook Download: PDF

2007 | 2007
XII, 290 Seiten
Springer New York (Verlag)
978-0-387-69469-6 (ISBN)

Infos

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.

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.

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Tensor algebra and tensor analysis were developed by Riemann, Christo?el, Ricci, Levi-Civita and others in the nineteenth century. The special theory of relativity, as propounded by Einstein in 1905, was elegantly expressed by Minkowski in terms of tensor ?elds in a ?at space-time. In 1915, Einstein formulated the general theory of relativity, in which the space-time manifold is curved. The theory is aesthetically and intellectually satisfying. The general theory of relativity involves tensor analysis in a pseudo- Riemannian manifold from the outset. Later, it was realized that even the pre-relativistic particle mechanics and continuum mechanics can be elegantly formulated in terms of tensor analysis in the three-dimensional Euclidean space. In recent decades, relativistic quantum ?eld theories, gauge ?eld theories, and various uni?ed ?eld theories have all used tensor algebra analysis exhaustively. This book develops from abstract tensor algebra to tensor analysis in va- ous di?erentiable manifolds in a mathematically rigorous and logically coherent manner. The material is intended mainly for students at the fourth-year and ?fth-year university levels and is appropriate for students majoring in either mathematical physics or applied mathematics.

Inhalt

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

Produkt-Details

• Erscheint lt. Verlag: 5.10.2007
• Zusatzinfo: XII, 290 p.
• Verlagsort: New York
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Biologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Allgemeines / Lexika
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Theoretische Physik
• Themenwelt: Technik
• Schlagworte: Calculus • Derivative • differential equation • Engineering • Relativity • Theoretical
• ISBN-10: 0-387-69469-2 / 0387694692
• ISBN-13: 978-0-387-69469-6 / 9780387694696