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Tensors

The Mathematics of Relativity Theory and Continuum Mechanics

  • Book
  • © 2007

Overview

Editors:
  1. Anadijiban Das

    1. Department of Mathematics and Pacific Institute for the Mathematical Sciences, Simon Fraser University, Burnaby, BC,Canada

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  • Many known concepts which are scattered in various books are brought together in a rigorous, logical way

  • Chapter 7 contains discussion on extrinsic curvature which is more extensive than in any other book available

  • Tensor analysis is further explained in the book, touching on general differential manifolds, manifolds with connections and manifolds with metrics and connections. Competing books have only Riemannian and Pseudo-Riemannian manifolds discussed

  • Each section of each chapter contains questions and exercises to further enhance understanding of the topics discussed

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Table of contents (7 chapters)

  1. Front Matter

    Pages I-XII
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  2. Finite-Dimensional Vector Spaces and Linear Mappings

    • Anadijiban Das
    Pages 1-15

  3. Tensor Algebra

    • Anadijiban Das
    Pages 16-51

  4. Tensor Analysis on a Differentiable Manifold

    • Anadijiban Das
    Pages 52-91

  5. Differentiable Manifolds with Connections

    • Anadijiban Das
    Pages 92-120

  6. Riemannian and Pseudo-Riemannian Manifolds

    • Anadijiban Das
    Pages 121-199

  7. Special Riemannian and Pseudo-Riemannian Manifolds

    • Anadijiban Das
    Pages 200-224

  8. Hypersurfaces, Submanifolds, and Extrinsic Curvature

    • Anadijiban Das
    Pages 225-256

  9. Back Matter

    Pages 257-289
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Keywords

About this book

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.

Reviews

From the reviews:

"This book is a very nice introduction to the theory of tensor analysis on differentiable manifolds. It is intended mainly for students, but it can also be useful to everyone interested in the tensor analysis on differentiable manifolds and its application to the relativity theory and continuum mechanics." (Cezar Dumitru Oniciuc, Zentralblatt MATH, Vol. 1138 (16), 2008)

Editors and Affiliations

  • Department of Mathematics and Pacific Institute for the Mathematical Sciences, Simon Fraser University, Burnaby, BC,Canada

    Anadijiban Das

About the editor

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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