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| The Einstein Summation Convention | |
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| Introduction | |
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| Repeated Indices in Sums | |
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| Double Sums | |
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| Substitutions | |
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| Kronecker Delta and Algebraic Manipulations | |
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| Basic Linear Algebra For Tensors | |
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| Introduction | |
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| Tensor Notation for Matrices, Vectors, and Determinants | |
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| Inverting a Matrix | |
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| Matrix Expressions for Linear Systems and Quadratic Forms | |
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| Linear Transformations | |
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| General Coordinate Transformations | |
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| The Chain Rule for Partial Derivatives | |
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| General Tensors | |
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| Coordinate Transformations | |
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| First-Order Tensors | |
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| Invariants | |
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| Higher-Order Tensors | |
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| The Stress Tensor | |
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| Cartesian Tensors | |
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| Tensor Operations: Tests For Tensor Character | |
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| Fundamental Operations | |
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| Tests for Tensor Character | |
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| Tensor Equations | |
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| The Metric Tensor | |
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| Introduction | |
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| Arc Length in Euclidean Space | |
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| Generalized Metrics; The Metric Tensor | |
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| Conjugate Metric Tensor; Raising and Lowering Indices | |
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| Generalized Inner-Product Spaces | |
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| Concepts of Length and Angle | |
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| The Derivative of a Tensor | |
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| Inadequacy of Ordinary Differentiation | |
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| Christoffel Symbols of the First Kind | |
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| Christoffel Symbols of the Second Kind | |
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| Covariant Differentiation | |
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| Absolute Differentiation along a Curve | |
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| Rules for Tensor Differentiation | |
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| Riemannian Geometry of Curves | |
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| Introduction | |
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| Length and Angle under an Indefinite Metric | |
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| Null Curves | |
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| Regular Curves: Unit Tangent Vector | |
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| Regular Curves: Unit Principal Normal and Curvature | |
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| Geodesics as Shortest Arcs | |
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| Riemannian Curvature | |
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| The Riemann Tensor | |
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| Properties of the Riemann Tensor | |
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| Riemannian Curvature | |
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| The Ricci Tensor | |
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| Spaces of Constant Curvature; Normal Coordinates | |
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| Zero Curvature and the Euclidean Metric | |
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| Flat Riemannian Spaces | |
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| Normal Coordinates | |
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| Schur's Theorem | |
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| The Einstein Tensor | |
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| Tensors in Euclidean Geometry | |
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| Introduction | |
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| Curve Theory; The Moving Frame | |
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| Curvature and Torsion | |
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| Regular Surfaces | |
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| Parametric Lines; Tangent Space | |
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| First Fundamental Form | |
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| Geodesics on a Surface | |
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| Second Fundamental Form | |
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| Structure Formulas for Surfaces | |
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| Isometries | |
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| Tensors in Classical Mechanics | |
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| Introduction | |
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| Particle Kinematics in Rectangular Coordinates | |
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| Particle Kinematics in Curvilinear Coordinates | |
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| Newton's Second Law in Curvilinear Coordinates | |
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| Divergence, Laplacian, Curl | |
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| Tensors in Special Relativity | |
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| Introduction | |
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| Event Space | |
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| The Lorentz Group and the Metric of SR | |
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| Simple Lorentz Matrices | |
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| Physical Implications of the Simple Lorentz Transformation | |
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| Relativistic Kinematics | |
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| Relativistic Mass, Force, and Energy | |
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| Maxwell's Equations in SR | |
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| Tensor Fields on Manifolds | |
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| Introduction | |
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| Abstract Vector Spaces and the Group Concept | |
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| Important Concepts for Vector Spaces | |
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| The Algebraic Dual of a Vector Space | |
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| Tensors on Vector Spaces | |
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| Theory of Manifolds | |
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| Tangent Space; Vector Fields on Manifolds | |
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| Tensor Fields on Manifolds | |
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| Answers to Supplementary Problems | |
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| Index | |