| The Einstein Summation Convention | p. 1 |
| Introduction | p. 1 |
| Repeated Indices in Sums | p. 1 |
| Double Sums | p. 2 |
| Substitutions | p. 2 |
| Kronecker Delta and Algebraic Manipulations | p. 3 |
| Basic Linear Algebra For Tensors | p. 8 |
| Introduction | p. 8 |
| Tensor Notation for Matrices, Vectors, and Determinants | p. 8 |
| Inverting a Matrix | p. 10 |
| Matrix Expressions for Linear Systems and Quadratic Forms | p. 10 |
| Linear Transformations | p. 11 |
| General Coordinate Transformations | p. 12 |
| The Chain Rule for Partial Derivatives | p. 13 |
| General Tensors | p. 23 |
| Coordinate Transformations | p. 23 |
| First-Order Tensors | p. 26 |
| Invariants | p. 28 |
| Higher-Order Tensors | p. 29 |
| The Stress Tensor | p. 29 |
| Cartesian Tensors | p. 31 |
| Tensor Operations: Tests For Tensor Character | p. 43 |
| Fundamental Operations | p. 43 |
| Tests for Tensor Character | p. 45 |
| Tensor Equations | p. 45 |
| The Metric Tensor | p. 51 |
| Introduction | p. 51 |
| Arc Length in Euclidean Space | p. 51 |
| Generalized Metrics; The Metric Tensor | p. 52 |
| Conjugate Metric Tensor; Raising and Lowering Indices | p. 55 |
| Generalized Inner-Product Spaces | p. 55 |
| Concepts of Length and Angle | p. 56 |
| The Derivative of a Tensor | p. 68 |
| Inadequacy of Ordinary Differentiation | p. 68 |
| Christoffel Symbols of the First Kind | p. 68 |
| Christoffel Symbols of the Second Kind | p. 70 |
| Covariant Differentiation | p. 71 |
| Absolute Differentiation along a Curve | p. 72 |
| Rules for Tensor Differentiation | p. 74 |
| Riemannian Geometry of Curves | p. 83 |
| Introduction | p. 83 |
| Length and Angle under an Indefinite Metric | p. 83 |
| Null Curves | p. 84 |
| Regular Curves: Unit Tangent Vector | p. 85 |
| Regular Curves: Unit Principal Normal and Curvature | p. 86 |
| Geodesics as Shortest Arcs | p. 88 |
| Riemannian Curvature | p. 101 |
| The Riemann Tensor | p. 101 |
| Properties of the Riemann Tensor | p. 101 |
| Riemannian Curvature | p. 103 |
| The Ricci Tensor | p. 105 |
| Spaces of Constant Curvature; Normal Coordinates | p. 114 |
| Zero Curvature and the Euclidean Metric | p. 114 |
| Flat Riemannian Spaces | p. 116 |
| Normal Coordinates | p. 117 |
| Schur's Theorem | p. 119 |
| The Einstein Tensor | p. 119 |
| Tensors in Euclidean Geometry | p. 127 |
| Introduction | p. 127 |
| Curve Theory; The Moving Frame | p. 127 |
| Curvature and Torsion | p. 130 |
| Regular Surfaces | p. 130 |
| Parametric Lines; Tangent Space | p. 132 |
| First Fundamental Form | p. 133 |
| Geodesics on a Surface | p. 135 |
| Second Fundamental Form | p. 136 |
| Structure Formulas for Surfaces | p. 137 |
| Isometries | p. 138 |
| Tensors in Classical Mechanics | p. 154 |
| Introduction | p. 154 |
| Particle Kinematics in Rectangular Coordinates | p. 154 |
| Particle Kinematics in Curvilinear Coordinates | p. 155 |
| Newton's Second Law in Curvilinear Coordinates | p. 156 |
| Divergence, Laplacian, Curl | p. 157 |
| Tensors in Special Relativity | p. 164 |
| Introduction | p. 164 |
| Event Space | p. 164 |
| The Lorentz Group and the Metric of SR | p. 166 |
| Simple Lorentz Matrices | p. 167 |
| Physical Implications of the Simple Lorentz Transformation | p. 169 |
| Relativistic Kinematics | p. 169 |
| Relativistic Mass, Force, and Energy | p. 171 |
| Maxwell's Equations in SR | p. 172 |
| Tensor Fields on Manifolds | p. 189 |
| Introduction | p. 189 |
| Abstract Vector Spaces and the Group Concept | p. 189 |
| Important Concepts for Vector Spaces | p. 190 |
| The Algebraic Dual of a Vector Space | p. 191 |
| Tensors on Vector Spaces | p. 193 |
| Theory of Manifolds | p. 194 |
| Tangent Space; Vector Fields on Manifolds | p. 197 |
| Tensor Fields on Manifolds | p. 199 |
| Answers to Supplementary Problems | p. 213 |
| Index | p. 223 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
