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| Vector Spaces | |
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| Definitions, properties, and examples | |
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| Representation of vector spaces | |
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| Linear mappings | |
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| Representation of linear mappings | |
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| Multilinear Mappings and Dual Spaces | |
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| Vector spaces of linear mappings | |
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| Vector spaces of multilinear mappings | |
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| Nondegenerate bilinear functions | |
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| Orthogonal complements and the transpose of a linear mapping | |
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| Tensor Product Spaces | |
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| The tensor product of two finite-dimensional vector spaces | |
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| Generalizations, isomorphisms, and a characterization | |
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| Tensor products of infinite-dimensional vector spaces | |
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| Tensors | |
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| Definitions and alternative interpretations | |
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| The components of tensors | |
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| Mappings of the spaces V[superscript r subscript s] | |
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| Symmetric and Skew-Symmetric Tensors | |
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| Symmetry and skew-symmetry | |
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| The symmetric subspace of V[superscript 0 subscript s] | |
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| The skew-symmetric (alternating) subspace of V[superscript 0 subscript s] | |
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| Some special properties of S[superscript 2](V*) and [Lambda superscript 2](V*) | |
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| Exterior (Grassmann) Algebra | |
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| Tensor algebras | |
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| Definition and properties of the exterior product | |
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| Some more properties of the exterior product | |
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| The Tangent Map of Real Cartesian Spaces | |
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| Maps of real cartesian spaces | |
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| The tangent and cotangent spaces at a point of R[superscript n] | |
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| The tangent map | |
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| Topological Spaces | |
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| Definitions, properties, and examples | |
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| Continuous mappings | |
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| Differentiable Manifolds | |
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| Definitions and examples | |
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| Mappings of differentiable manifolds | |
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| The tangent and cotangent spaces at a point of M | |
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| Some properties of mappings | |
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| Submanifolds | |
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| Parametrized submanifolds | |
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| Differentiable varieties as submanifolds | |
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| Vector Fields, 1-Forms, and Other Tensor Fields | |
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| Vector fields | |
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| 1-Form fields | |
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| Tensor fields and differential forms | |
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| Mappings of tensor fields and differential forms | |
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| Differentiation and Intergration of Differential Forms | |
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| Exterior differentiation of differential forms | |
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| Integration of differential forms | |
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| The Flow and the Lie Derivative of A Vector Field | |
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| Integral curves and the flow of a vector field | |
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| Flow boxes (local flows) and complete vector fields | |
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| Coordinate vector fields | |
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| The Lie derivative | |
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| Integrability Conditions for Distributions and for Pfaffian Systems | |
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| Completely integrable distributions | |
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| Completely integrable Pfaffian systems | |
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| The characteristic distribution of a differential system | |
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| Pseudo-Riemannian Geometry | |
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| Pseudo-Riemannian manifolds | |
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| Length and distance | |
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| Flat spaces | |
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| Connection 1-Forms | |
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| The Levi-Civita connection and its covariant derivative | |
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| Geodesics of the Levi-Civita connection | |
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| The torsion and curvature of a linear, or affine connection | |
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| The exponential map and normal coordinates | |
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| Connections on pseudo-Riemannian manifolds | |
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| Connections on Manifolds | |
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| Connections between tangent spaces | |
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| Coordinate-free description of a connection | |
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| The torsion and curvature of a connection | |
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| Some geometry of submanifolds | |
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| Mechanics | |
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| Symplectic forms, symplectic mappings, Hamiltonian vector fields, and Poisson brackets | |
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| The Darboux theorem, and the natural symplectic structure of T* M | |
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| Hamilton's equations. Examples of mechanical systems | |
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| The Legendre transformation and Lagrangian vector fields | |
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| Additional Topics in Mechanics | |
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| The configuration space as a pseudo-Riemannian manifold | |
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| The momentum mapping and Noether's theorem | |
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| Hamilton-Jacobi theory | |
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| A Spacetime | |
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| Newton's mechanics and Maxwell's electromagnetic theory | |
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| Frames of reference generalized | |
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| The Lorentz transformations | |
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| Some properties and forms of the Lorentz transformations | |
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| Minkowski spacetime | |
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| Some Physics on Minkowski Spacetime | |
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| Time dilation and the Lorentz-Fitzgerald contraction | |
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| Particle dynamics on Minkowski spacetime | |
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| Electromagnetism on Minkowski spacetime | |
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| Perfect fluids on Minkowski spacetime | |
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| Einstein Spacetimes | |
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| Gravity, acceleration, and geodesics | |
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| Gravity is a manifestation of curvature | |
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| The field equation in empty space | |
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| Einstein's field equation (Sitz, der Preuss Acad. Wissen., 1917) | |
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| Spacetimes Near an Isolated Star | |
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| Schwarzschild's exterior solution | |
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| Two applications of Schwarzschild's solution | |
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| The Kruskal extension of Schwarzschild spacetime | |
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| The field of a rotating star | |
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| Nonempty Spacetimes | |
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| Schwarzschild's interior solution | |
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| The form of the Friedmann-Robertson-Walker metric tensor and its properties | |
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| Friedmann-Robertson-Walker spacetimes | |
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| Lie Groups | |
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| Definition and examples | |
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| Vector fields on a Lie group | |
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| Differential forms on a Lie group | |
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| The action of a Lie group on a manifold | |
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| Fiber Bundles | |
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| Principal fiber bundles | |
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| Examples | |
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| Associated bundles | |
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| Examples of associated bundles | |
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| Connections on Fiber Bundles | |
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| Connections on principal fiber bundles | |
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| Curvature | |
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| Linear Connections | |
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| Connections on vector bundles | |
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| Gauge Theory | |
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| Gauge transformation of a principal bundle | |
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| Gauge transformations of a vector bundle | |
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| How fiber bundles with connections form the basic framework of the Standard Model of elementary particle physics | |
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| References | |
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| Notation | |
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| Index | |