An Introduction to Tensor Analysis

The subject of Tensor Analysis deals with the problem of the formulation of the relation between various entities in forms which remain invariant when we pass from one system of coordinates to another. The invariant form of equation is necessarily related to the possible system of coordinates with reference to which the equation remains invariant. The primary purpose of this book is the study of the invariance form of equation relative to the totally of the rectangular co-ordinate system in the three-dimensional Euclidean space. We start with the consideration of the way the sets representing various entities are transformed when we pass from one system of rectangular co-ordinates to another. A Tensor may be a physical entity that can be described as a Tensor only with respect to the manner of its representation by means of multi-sux sets associated with different system of axes such that the sets associated with different system of co-ordinate obey the transformation law for Tensor. We have employed sux notation for tensors of any order, we could also employ single letter such A,B to denote Tensors.

Preface v -- Syllabus vii -- 1 Introduction 1 -- 1.1 Symbols Multi-Suffix 2 -- 1.2 Summation Convention 3 -- 2 Cartesian Tensor 7 -- 2.1 Introduction 7 -- 2.2 Transformation of Coordinates 8 -- 2.3 Relations Between the Direction Cosines 10 -- 2.4 Transformation of Velocity Components 11 -- 2.5 First-Order Tensors 12 -- 2.6 Second-Order Tensors 13 -- 2.7 Notation for Tensors 14 -- 2.8 Algebraic Operations on Tensors 14 -- 2.8.1 Sum and Difference of Tensors 15 -- 2.8.2 Product of Tensors 16 -- 2.9 Quotient Law of Tensors 17 -- 2.10 Contraction Theorem 19 -- 2.11 Symmetric and Skew-Symmetric Tensor 21 -- 2.12 Alternate Tensor 23 -- 2.13 Kronecker Tensor 24 -- 2.14 Relation Between Alternate and Kronecker Tensors 25 -- 2.15 Matrices and Tensors of First and Second Orders 26 -- 2.16 Product of Two Matrices 28 -- 2.17 Scalar and Vector Inner Product 31 -- 2.17.1 Two Vectors 31 -- 2.17.2 Scalar Product 31 -- 2.17.3 Vector Product 31 -- 2.18 Tensor Fields 32 -- 2.18.1 Gradient of Tensor Field 32 -- 2.18.2 Divergence of Vector Point Function 34 -- 2.18.3 Curl of Vector Point Function 34 -- 2.19 Tensorial Formulation of Gauss's Theorem 35 -- 2.20 Tensorial Formulation of Stoke's Theorem 3…