Introduction to Differential Geometry with Tensor Applications. Группа авторов

      

Table of Contents

      1  Cover

      2  Title Page

      3  Copyright

      4  Dedication

      5  Preface

      6  About the Book

      7  Introduction

      8  Part I: Tensor Theory 1 Preliminaries 1.1 Introduction 1.2 Systems of Different Orders 1.3 Summation Convention Certain Index 1.4 Kronecker Symbols 1.5 Linear Equations 1.6 Results on Matrices and Determinants of Systems 1.7 Differentiation of a Determinant 1.8 Examples 1.9 Exercises 2 Tensor Algebra 2.1 Introduction 2.2 Scope of Tensor Analysis 2.3 Transformation of Coordinates in Sn 2.4 Transformation by Invariance 2.5 Transformation by Covariant Tensor and Contravariant Tensor 2.6 The Tensor Concept: Contravariant and Covariant Tensors 2.7 Algebra of Tensors 2.8 Symmetric and Skew-Symmetric Tensors 2.9 Outer Multiplication and Contraction 2.10 Quotient Law of Tensors 2.11 Reciprocal Tensor of a Tensor 2.12 Relative Tensor, Cartesian Tensor, Affine Tensor, and Isotropic Tensors 2.13 Examples 2.14 Exercises 3 Riemannian Metric 3.1 Introduction 3.2 The Metric Tensor 3.3 Conjugate Tensor 3.4 Associated Tensors 3.5 Length of a Vector 3.6 Angle Between Two Vectors 3.7 Hypersurface 3.8 Angle Between Two Coordinate Hypersurfaces 3.9 Exercises 4 Tensor Calculus 4.1 Introduction 4.2 Christoffel Symbols 4.3 Transformation of Christoffel Symbols 4.4 Covariant Differentiation of Tensor 4.5 Gradient, Divergence, and Curl 4.6 Exercises 5 Riemannian Geometry 5.1 Introduction 5.2 Riemannian-Christoffel Tensor 5.3 Properties of Riemann-Christoffel Tensors 5.4 Ricci Tensor, Bianchi Identities, Einstein Tensors 5.5 Einstein Space 5.6 Riemannian and Euclidean Spaces 5.7 Exercises 6 The e-Systems and the Generalized Kronecker Deltas 6.1 Introduction 6.2 e-Systems 6.3 Generalized Kronecker Delta 6.4 Contraction of

6.5 Application of e-Systems to Determinants and Tensor Characters of Generalized Kronecker Deltas 6.6 Exercises

      9  Part II: Differential Geometry 7 Curvilinear Coordinates in Space 7.1 Introduction 7.2 Length of Arc 7.3 Curvilinear Coordinates in E₃ 7.4 Reciprocal Base Systems 7.5 Partial Derivative 7.6 Exercises 8 Curves in Space 8.1 Introduction 8.2 Intrinsic Differentiation 8.3 Parallel Vector Fields 8.4 Geometry of Space Curves