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

This preview shows page 3 - 8 out of 171 pages.

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
Image of page 3
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
Image of page 4
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
Image of page 5
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
Image of page 6
List of Figures 1 Cartesian coordinate system and its basis vectors with components of a vector 17 2
Image of page 7
Image of page 8

You've reached the end of your free preview.

Want to read all 171 pages?

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

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask ( soon) You can ask (will expire )
Answers in as fast as 15 minutes
A+ icon
Ask Expert Tutors