(The principal aim of tensor analysis is to investigate th...)
The principal aim of tensor analysis is to investigate the relations which remain valid when we change from one coordinate system to another. Albert Einstein found it to be an excellent tool for the presentation of his general theory of relativity and consequently tensor analysis came to prominence in mathematics.
It has applications in most branches of theoretical physics and engineering. This present book is intended as a text for postgraduate students of mathematics, physics and engineering. It is self-contained and requires prior knowledge of elementary calculus, differential equations and classical mechanics. It consists of five chapters, each containing a large number of solved examples, unsolved problems and links to the solution of these problems.
'Tensor Analysis with Applications' can be used on a selection of university courses, and will be a welcome addition to the library of maths, physics and engineering departments.
Contents
1. Tensors and their Algebra 1.1 Introduction 1.2 Transformation of Coordinates 1.3 Summation Convention 1.4 Kronecker Delta 1.5 Scalars, Contravariant and Covariant Vectors 1.6 Tensors of Higher Rank 1.7 Symmetry of Tensors 1.8 Algebra of Tensors, Addition and Subtraction, Equality of Tensors, Inner and Outer Products, Contraction, The Quotient Law 1.9 Irreducible Tensors, Exercises 2. Riemannian Space and Metric Tensor 2.1 Introduction 2.2 The Metric Tensor 2.3 Raising and Lowering of Indices-Associated Tensor 2.4 Vector Magnitude 2.5 Relative and Absolute Tensors 2.6 The Levi-Civita Tensor, Exercises 3. Christoffel Symbols and Covariant Differentiation 3.1 Introduction 3.2 Christoffel Symbols 3.3 Transformation Law for Christoffel Symbols 3.4 Equation of a Geodesic 3.5 Affine Parameter 3.6 Geodesic Coordinate System 3.7 Covariant Differentiation, Covariant Derivatives of Contravariant and Covariant Vectors, Covariant Derivatives of Rank Two Tensors, Covariant Derivatives of Tensors of Higher Rank 3.8 Rules for Covariant Differentiation 3.9 Some Useful Formulas, Divergence of a Vector Field, Gradient of a Scalar and Laplacian, Curl of a Vector Field, Divergence of a Tensor Field 3.10 Intrinsic Derivative-Parallel Transport 3.11 Null Geodesics 3.12 Alternative Derivation of Equation of Geodesic, Exercises 4. The Riemann Curvature Tensor 4.1 Introduction 4.2 The Riemann Curvature Tensor 4.3 Commutation of Covariant Derivatives 4.4 Covariant Form of the Riemann Curvature Tensor 4.5 Properties of the Riemann Curvature Tensor 4.6 Uniqueness of the Riemann Curvature Tensor 4.7 The Number of Algebraically Independent Components of the Riemann Curvature Tensor 4.8 The Ricci Tensor and the Scalar Curvature 4.9 The Einstein Tensor 4.10 The Integrability of the Riemann Tensor and the Flatness of the Space 4.11 The Einstein Spaces 4.12 Curvature of a Riemannian Space 4.13 Spaces of Constant Curvature, Exercises 5. Some Advanced Topics 5.1 Introduction, 5.2 Gewodesic Deviation 5.3 Decomposition of Riemann Curvature Tensor 5.4 Electric and Magnetic Parts of the Riemann and Weyl Tensors 5.5 Classification of Gravitational Fields 5.6 Invariants of the Riemann Curvature Tensor 5.7 Curvature Tensors Involving the Riemann Tensor, Space-matter Tensor, Conharmonic Curvature Tensor 5.8 Lie Derivative 5.9 The Killing Equation 5.10 The Curvature Tensor and Killing Vector, Exercises
Also available:
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Zafar Ahsan, Indian mathematics professor. Achievements include research in general relativity & gravitation, Quran & modern cosmology.
Background
Ahsan, Zafar was born on March 11, 1951 in Mainpuri, Uttar Pradesh, India. Son of Mohammad Hameed Khan and Aziz Fatima.
Education
Master of Science, Aligarh Muslim University, Uttar Pradesh. Master of Philosophy, Aligarh Muslim University, Uttar Pradesh. Doctor of Philosophy, Aligarh Muslim University, Uttar Pradesh.
Career
Lecturer mathematics Aligarh Muslim University, 1978—1992, reader mathematics, 1992—2000, professor mathematics, since 2000.
Contributor articles to professional research journals. Author: (book) Differential Equations & Their Applications, 1999, Vector Analysis, 2003, Mathematical Methods, 2005, Tensor Analysis with Applications, 2008.
Membership
Member of J. California Math Society, Alig Bull Mathematics (managing director), American Biography Institute (research board advisors), Astronomical Society India, Calcutta Mathematics Society, International Egyptian Engineering Mathematics Society (founder), Indian Society Industrial & Applied Mathematics, Tensor Society India (editor-in-chief), Indian Mathematics Society, Indian Association General Relativity & Gravitation (life. Council member 2002-2006).
Connections
Married Nikhat Malik, April 6, 1979. Children: Mohammad Khawer, Mohammad Haris Khan.
Recipient Visiting Associate award, Inter Unive...Recipient Visiting Associate award, Inter University Center Astronomy & Astrophysics, Pune, India, 1996—2005, Rashtriya Gaurav award, 2007, Shiksha Rattan Purashkar, 2007, Best Citizen of India award, 2006-2008. Junior & Senior Research fellowship, Council of Scientific & Industrial Research, India, 1974-1978, University Grants Commission Visiting fellow, Sardar Patel University, Vallabh Vidyanagar, India, 2004-2005.