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| Introduction | |
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| Tensor Algebra | |
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| Notation and Systems of Numbers | |
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| Introduction and Basic Concepts | |
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| Symmetric and Antisymmetric Systems | |
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| Operations with Systems | |
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| Addition and Subtraction of Systems | |
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| Direct Product of Systems | |
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| Contraction of Systems | |
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| Composition of Systems | |
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| Summation Convention | |
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| Unit Symmetric and Antisymmetric Systems 3 Vector Spaces | |
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| Introduction and Basic Concepts | |
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| Defnition of a Vector Space | |
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| The Euclidean Metric Space | |
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| The Riemannian Spaces 4 Definitions of Tensors | |
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| Transformations of Variables | |
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| Contravariant Vectors | |
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| Covariant Vectors | |
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| Invariants (Scalars) | |
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| Contravariant Tensors | |
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| Covariant Tensors | |
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| Mixed Tensors | |
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| Symmetry Properties of Tensors | |
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| Symmetric and Antisymmetric Parts of Tensors | |
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| Tensor Character of Systems 5 Relative Tensors | |
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| Introduction and Definitions | |
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| Unit Antisymmetric Tensors | |
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| Vector Product in Three Dimensions | |
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| Mixed Product in Three Dimensions | |
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| Orthogonal Coordinate Transformations | |
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| Rotations of Descartes Coordinates | |
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| Translations of Descartes Coordinates | |
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| Inversions of Descartes Coordinates | |
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| Axial Vectors and Pseudoscalars in Descartes Coordinates 6 The Metric Tensor | |
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| Introduction and Definitions | |
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| Associated Vectors and Tensors | |
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| Arc Length of Curves. Unit Vectors | |
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| Angles between Vectors | |
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| Schwarz Inequality | |
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| Orthogonal and Physical Vector Coordinates 7 Tensors as Linear Operators | |
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| Tensor Analysis | |
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| Tensor Derivatives | |
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| Differentials of Tensors | |
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| Differentials of Contravariant Vectors | |
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| Differentials of Covariant Vectors | |
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| Covariant Derivatives | |
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| Covariant Derivatives of Vectors | |
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| Covariant Derivatives of Tensors | |
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| Properties of Covariant Derivatives | |
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| Absolute Derivatives of Tensors | |
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| Christoffel Symbols | |
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| Properties of Christoff Symbols | |
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| Relation to the Metric Tensor | |
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| Differential Operators | |
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| The Hamiltonian r-Operator | |
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| Gradient of Scalars | |
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| Divergence of Vectors and Tensors | |
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| Curl of Vectors | |
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| Laplacian of Scalars and Tensors | |
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| Integral Theorems for Tensor Fields | |
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| Stokes Theorem | |
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| Gauss Theorem | |
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| Geodesic Lines | |
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| Lagrange Equations | |
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| Geodesic Equations | |
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| The Curvature Tensor | |
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| Definition of the Curvature Tensor | |
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| Properties of the Curvature Tensor | |
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| Commutator of Covariant Derivatives | |
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| Ricci Tensor and Scalar | |
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| Curvature Tensor Components | |
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| Special Theory of Relativity | |
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| Relativistic Kinematics | |
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| The Principle of Relativity | |
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| Invariance of the Speed of Light | |
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| The Interval between Events | |
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| Lorentz Transformations | |
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| Velocity and Acceleration Vectors | |
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| Relativistic Dynamics | |
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| Lagrange Equations | |
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| Energy-Momentum Vector | |
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| Introduction and Definitions | |
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| Transformations of Energy-Momentum | |
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| Conservation of Energy-Momentum | |
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| Angular Momentum Tensor | |
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| Electromagnetic Fields | |
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| Electromagnetic Field Tensor | |
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| Gauge Invariance | |
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| Lorentz Transformations and Invariants | |
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| Electromagnetic Field Equations | |
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| Electromagnetic Current Vector | |
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| Maxwell Equations | |
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| Electromagnetic Potentials | |
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| Energy-Momentum Tensor | |
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| General Theory of Relativity | |
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| Gravitational Fields | |
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| Introduction | |
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| Time Intervals and Distances | |
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| Particle Dynamics | |
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| Electromagnetic Field Equations | |
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| Gravitational Field Equations | |
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| The Action Integral | |
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| Action for Matter Fields | |
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| Einstein Field Equations | |
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| Solutions of Field Equations | |
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| The Newton Law | |
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| The Schw | |