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Table of Contents
Intro
Foreword
Preface to the English Edition
Preface to the Second Edition
Preface to the First Edition
Contents
Outline of Volume II
Outline of Volume III
1 Topological Spaces in Brief
1.1 The ABCs of Set Theory
1.2 Topological Spaces
1.3 Compactness [Optional Reading]
Exercises
Reference
2 Manifolds and Tensor Fields
2.1 Differentiable Manifolds
2.2 Tangent Vectors and Tangent Vector Fields
2.2.1 Tangent Vectors
2.2.2 Tangent Vector Fields on Manifolds
2.3 Dual Vector Fields
2.4 Tensor Fields
2.5 Metric Tensor Fields
2.6 The Abstract Index Notation
Exercises
References
3 The Riemann (Intrinsic) Curvature Tensor
3.1 Derivative Operators
3.2 Derivative and Parallel Transport of a Vector Field Along a Curve
3.2.1 Parallel Transport of a Vector Field Along a Curve
3.2.2 The Derivative Operator Associated with a Metric
3.2.3 Relationship Between the Derivative and Parallel Transport of a Vector Field Along a Curve
3.3 Geodesics
3.4 The Riemann Curvature Tensor
3.4.1 Definition and Properties of the Riemann Curvature
3.4.2 Computing Riemann Curvature from a Metric
3.5 The Intrinsic Curvature and the Extrinsic Curvature
Exercises
References
4 Lie Derivatives, Killing Fields and Hypersurfaces
4.1 Maps of Manifolds
4.2 Lie Derivatives
4.3 Killing Vector Fields
4.4 Hypersurfaces
Exercises
References
5 Differential Forms and Their Integrals
5.1 Differential Forms
5.2 Integration on Manifolds
5.3 Stokes's Theorem
5.4 Volume Elements
5.5 Integrating Functions on Manifolds, Gauss's Theorem
5.6 Dual Differential Forms
5.7 Computing the Riemann Curvature Using the Tetrad Method [Optional Reading]
Exercises
6.4 The Energy-Momentum Tensor of Continuous Media
6.5 Perfect Fluid Dynamics
6.6 Electrodynamics
6.6.1 Electromagnetic Fields and 4-Current Densities
6.6.2 Maxwell's Equations
6.6.3 Lorentz 4-Force
6.6.4 The Energy-Momentum Tensor of an Electromagnetic Field
6.6.5 Electromagnetic 4-Potential and Its Equation of Motion, Electromagnetic Waves
6.6.6 The Doppler Effect on a Light Wave
Exercises
References
7 Foundations of General Relativity
7.1 Gravity and Spacetime Geometry
7.2 Physical Laws in Curved Spacetime
Foreword
Preface to the English Edition
Preface to the Second Edition
Preface to the First Edition
Contents
Outline of Volume II
Outline of Volume III
1 Topological Spaces in Brief
1.1 The ABCs of Set Theory
1.2 Topological Spaces
1.3 Compactness [Optional Reading]
Exercises
Reference
2 Manifolds and Tensor Fields
2.1 Differentiable Manifolds
2.2 Tangent Vectors and Tangent Vector Fields
2.2.1 Tangent Vectors
2.2.2 Tangent Vector Fields on Manifolds
2.3 Dual Vector Fields
2.4 Tensor Fields
2.5 Metric Tensor Fields
2.6 The Abstract Index Notation
Exercises
References
3 The Riemann (Intrinsic) Curvature Tensor
3.1 Derivative Operators
3.2 Derivative and Parallel Transport of a Vector Field Along a Curve
3.2.1 Parallel Transport of a Vector Field Along a Curve
3.2.2 The Derivative Operator Associated with a Metric
3.2.3 Relationship Between the Derivative and Parallel Transport of a Vector Field Along a Curve
3.3 Geodesics
3.4 The Riemann Curvature Tensor
3.4.1 Definition and Properties of the Riemann Curvature
3.4.2 Computing Riemann Curvature from a Metric
3.5 The Intrinsic Curvature and the Extrinsic Curvature
Exercises
References
4 Lie Derivatives, Killing Fields and Hypersurfaces
4.1 Maps of Manifolds
4.2 Lie Derivatives
4.3 Killing Vector Fields
4.4 Hypersurfaces
Exercises
References
5 Differential Forms and Their Integrals
5.1 Differential Forms
5.2 Integration on Manifolds
5.3 Stokes's Theorem
5.4 Volume Elements
5.5 Integrating Functions on Manifolds, Gauss's Theorem
5.6 Dual Differential Forms
5.7 Computing the Riemann Curvature Using the Tetrad Method [Optional Reading]
Exercises
6.4 The Energy-Momentum Tensor of Continuous Media
6.5 Perfect Fluid Dynamics
6.6 Electrodynamics
6.6.1 Electromagnetic Fields and 4-Current Densities
6.6.2 Maxwell's Equations
6.6.3 Lorentz 4-Force
6.6.4 The Energy-Momentum Tensor of an Electromagnetic Field
6.6.5 Electromagnetic 4-Potential and Its Equation of Motion, Electromagnetic Waves
6.6.6 The Doppler Effect on a Light Wave
Exercises
References
7 Foundations of General Relativity
7.1 Gravity and Spacetime Geometry
7.2 Physical Laws in Curved Spacetime