In view of all the above factors the present text can be used as a textbook or

# In view of all the above factors the present text can

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In view of all the above factors, the present text can be used as a textbook or as a reference for an introductory course on tensor algebra and calculus or as a guide for self- studying and learning. I tried to be as clear as possible and to highlight the key issues of the subject at an introductory level in a concise form. I hope I have achieved some success in reaching these objectives for the majority of my target audience. Finally, I should make a short statement about credits in making this book following the tradition in writing book prefaces. In fact everything in the book is made by the author including all the graphic illustrations, front and back covers, indexing, typesetting, and overall design. However, I should acknowledge the use of the L A T E X typesetting package and the L A T E X based document preparation package L Y X for facilitating many things in typesetting and design which cannot be done easily or at all without their versatile and powerful capabilities. I also used the Ipe extensible drawing editor program for making all the graphic illustrations in the book as well as the front and back covers. Taha Sochi London, November 2016 2
Contents Preface 1 Table of Contents 3 List of Figures 6 Nomenclature 7 1 Preliminaries 8 1.1 Historical Overview of Development & Use of Tensor Calculus . . . . . . . 8 1.2 General Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 General Mathematical Background . . . . . . . . . . . . . . . . . . . . . . 15 1.3.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.3.2 Vector Algebra and Calculus . . . . . . . . . . . . . . . . . . . . . . 23 1.3.3 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2 Tensors 47 2.1 General Background about Tensors . . . . . . . . . . . . . . . . . . . . . . 47 2.2 General Terms and Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3 General Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.4 Examples of Tensors of Different Ranks . . . . . . . . . . . . . . . . . . . . 57 2.5 Applications of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.6 Types of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.6.1 Covariant and Contravariant Tensors . . . . . . . . . . . . . . . . . 59 2.6.2 True and Pseudo Tensors . . . . . . . . . . . . . . . . . . . . . . . . 65 3
2.6.3 Absolute and Relative Tensors . . . . . . . . . . . . . . . . . . . . . 68 2.6.4 Isotropic and Anisotropic Tensors . . . . . . . . . . . . . . . . . . . 70 2.6.5 Symmetric and Anti-symmetric Tensors . . . . . . . . . . . . . . . . 70 2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3 Tensor Operations 83 3.1 Addition and Subtraction . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2 Multiplication of Tensor by Scalar . . . . . . . . . . . . . . . . . . . . . . . 84 3.3 Tensor Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.4 Contraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 3.5 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 3.6 Permutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.7 Tensor Test: Quotient Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 91 3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4 δ and Tensors 96 4.1 Kronecker δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.2 Permutation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3 Useful Identities Involving δ or/and . . . . . . . . . . . . . . . . . . . . . 101 4.3.1 Identities Involving δ . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.3.2 Identities Involving . . . . . . . . . . . . . . . . . . . . . . . . . . 104 4.3.3 Identities Involving δ and . . . . . . . . . . . . . . . . . . . . . . . 106 4.4 Generalized Kronecker δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5 Applications of Tensor Notation and Techniques 116 5.1 Common Definitions in Tensor Notation . . . . . . . . . . . . . . . . . . . 116 5.2 Scalar Invariants of Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . 119 4
5.3 Common Differential Operations in Tensor Notation . . . . . . . . . . . . . 120 5.3.1 Cartesian Coordinate System . . . . . . . . . . . . . . . . . . . . . 121 5.3.2 Cylindrical Coordinate System . . . . . . . . . . . . . . . . . . . . . 124 5.3.3 Spherical Coordinate System . . . . . . . . . . . . . . . . . . . . . . 125 5.3.4 General Orthogonal Coordinate System . . . . . . . . . . . . . . . . 126 5.4 Common Identities in Vector and Tensor Notation . . . . . . . . . . . . . . 127 5.5 Integral Theorems in Tensor Notation . . . . . . . . . . . . . . . . . . . . . 132 5.6 Examples of Using Tensor Techniques to Prove Identities . . . . . . . . . . 134 5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 6 Metric Tensor 149 6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 7 Covariant Differentiation 155 7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 References 162 Index 163 5
List of Figures 1 Cartesian coordinate system and its basis vectors with components of a vector 17 2

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• Summer '20
• Rajendra Paramanik
• Tensor, Coordinate system, Polar coordinate system, Coordinate systems

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