TUM Library

Cover; Half Title; Series Page; Title Page; Copyright Page; Contents; Preface; About the Book; Author; Part I: Formalism of Tensor Calculus; 1. Prerequisites for Tensors; 1.1 Ideas of Coordinate Systems; 1.2 Curvilinear Coordinates and Contravariant and Covariant Components of a Vector (the Entity); 1.3 Quadratic Forms, Properties, and Classifications; 1.4 Quadratic Differential Forms and Metric of a Space in the Form of Quadratic Differentials; Exercises; 2. Concept of Tensors; 2.1 Some Useful Definitions; 2.2 Transformation of Coordinates; 2.3 Second and Higher Order Tensors

2.4 Operations on Tensors2.5 Symmetric and Antisymmetric (or Skew-Symmetric) Tensors; 2.6 Quotient Law; Exercises; 3. Riemannian Metric and Fundamental Tensors; 3.1 Riemannian Metric; 3.2 Cartesian Coordinate System and Orthogonal Coordinate System; 3.3 Euclidean Space of n Dimensions, Euclidean Co-Ordinates, and Euclidean Geometry; 3.4 The Metric Functions g[sub(ij)] Are Second-Order Covariant Symmetric Tensors; 3.5 The Function g[sub(ij)] Is a Contravariant Second-Order Symmetric Tensor; 3.6 Scalar Product and Magnitude of Vectors; 3.7 Angle Between Two Vectors and Orthogonal Condition

4.7 Covariant Derivative of Contravariant Tensor of Rank One4.8 Covariant Derivative of Covariant Tensor of Rank Two; 4.9 Covariant Derivative of Contravariant Tensor of Rank Two; 4.10 Covariant Derivative of Mixed Tensor of Rank Two; 4.10.1 Generalization; 4.11 Covariant Derivatives of g[sub(ij)] g[sup(ij)] and also g[sub(i)][sub(j)]; 4.12 Covariant Differentiations of Sum (or Difference) and Product of Tensors; 4.13 Gradient of an Invariant Function; 4.14 Curl of a Vector; 4.15 Divergence of a Vector; 4.16 Laplacian of a Scalar Invariant; 4.17 Intrinsic Derivative or Derived Vector of v

4.18 Definition: Parallel Displacement of Vectors4.18.1 When Magnitude Is Constant; 4.18.2 Parallel Displacement When a Vector Is of Variable Magnitude; Exercises; 5. Properties of Curves in V[sub(n)] and Geodesics; 5.1 The First Curvature of a Curve; 5.2 Geodesics; 5.3 Derivation of Differential Equations of Geodesics; 5.4 Aliter: Differential Equations of Geodesics as Stationary Length; 5.5 Geodesic Is an Autoparallel Curve; 5.6 Integral Curve of Geodesic Equations; 5.7 Riemannian and Geodesic Coordinates, and Conditions for Riemannian and Geodesic Coordinates

The aim of this book is to make the subject easier to understand. This book provides clear concepts, tools, and techniques to master the subject -tensor, and can be used in many fields of research. Special applications are discussed in the book, to remove any confusion, and for absolute understanding of the subject. In most books, they emphasize only the theoretical development, but not the methods of presentation, to develop concepts. Without knowing how to change the dummy indices, or the real indices, the concept cannot be understood. This book takes it down a notch and simplifies the topic for easy comprehension. Features Provides a clear indication and understanding of the subject on how to change indices Describes the original evolution of symbols necessary for tensors Offers a pictorial representation of referential systems required for different kinds of tensors for physical problems Presents the correlation between critical concepts Covers general operations and concepts

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