| Description |
1 online resource |
| Contents |
Cover; Half Title; Title Page; Copyright Page; Contents; Preface; 1 INTRODUCTION; Space curves; Curves on surfaces; 2 SMOOTH MANIFOLDS AND VECTOR BUNDLES; Smooth manifolds; Smooth maps; Partitions of unity; The rank of a map; Submanifolds; Vector bundles; The tangent bundle; 3 VECTOR FIELDS AND DIFFERENTIAL EQUATIONS; Vector fields; Integral curves and local flows; Global flows; 4 TENSORS; Dual vector bundles; Tensor bundles; Contraction; The Lie derivative; Distributions and integral manifolds; 5 DIFFERENTIAL FORMS; Exterior forms on a vector space; Differential forms on a manifold |
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Exterior differentiation of formsWedge product of vector-valued forms; Interior product; The Lie derivative of forms; 6 INTEGRATION ON MANIFOLDS; Manifolds with boundary; Exact forms; Orientation; Integration of differential forms; 7 METRIC AND SYMPLECTIC STRUCTURES; Covariant tensors of degree 2; Pseudo-Riemannian manifolds; The Hodge star operator; Time dependent vector fields; Symplectic manifolds; Hamiltonian systems; Lagrangian systems; Conservative systems; Time dependent systems; 8 LIE GROUPS; Lie groups and their Lie algebras; Group representations; Lie subgroups; Coverings |
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The exponential mapClosed subgroups; Matrix groups; The algebra of quaternions; Left invariant forms; 9 GROUP ACTIONS; Introduction; The adjoint representation; The groups SO(3) and S[sup(3)]; The Lorentz group and Sl(2,C); Semidirect products; Affine spaces; Infinitesimal group actions; Hamiltonian systems with symmetry; Lagrangian systems with symmetry; Gravitational central fields; 10 FIBRE BUNDLES; Introduction; Induced bundles; Principal fibre bundles; Associated bundles; Connections; Tensorial forms; Covariant derivative of forms on principal bundles; The curvature form |
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Horizontal lifts of vector fieldsLocal sections and trivializations; Horizontal lifts of curves; Parallel transport; Forms in associated bundles; Covariant derivative of sections in associated vector bundles; Covariant derivative of tensor fields; Covariant derivative of sections along smooth maps; Linear connections; Koszul connections; Structure equations; Geodesics; Metrical connections; The Schwarzschild -- de Sitter spacetime; Affine transformations and Killing vector fields; Conformal transformations; 11 ISOMETRIC IMMERSIONS AND THE SECOND FUNDAMENTAL FORM |
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Connections in reduced subbundlesThe normal bundle and the bundle of adapted orthonormal frames; The second fundamental form; The shape tensor; The shape operator; The formulae of Gauss and Weingarten; Strain and vorticity; The equations of Gauss, Ricci and Codazzi; Pseudo-Riemannian hypersurfaces; The Robertson-Walker spacetime; The Friedmann cosmological models; 12 JET BUNDLES; Bundles; Affine bundles; Derivations and the Frölicher-Nijenhuis bracket; First order jet bundles; Holonomic tangent vectors; Contact cotangent vectors; Jet fields and connections; Equivariant jet fields |
| Summary |
This text provides a systematic presentation of the mathematical foundation of modern physics with applications particularly within classical mechanics and the theory of relativity. This book provides a systematic presentation of the mathematical foundation of modern physics with applications particularly within classical mechanics and the theory of relativity. Among the themes presented in the book are differentiable manifolds, differential forms, fiber bundles and differential geometry with non-trivial applications especially within the general theory of relativity. The emphasis is upon a systematic and logical construction of the mathematical foundations. The book is self contained with definitions, theorems and detailed proofs. The book will also appeal to students of theoretical physics interested in the mathematical foundation of the theories |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Print version record |
| Subject |
Manifolds (Mathematics)
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Fiber bundles (Mathematics)
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Differential equations.
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Lie groups.
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MATHEMATICS -- Topology.
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Differential equations
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Fiber bundles (Mathematics)
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Lie groups
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Manifolds (Mathematics)
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| Form |
Electronic book
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| ISBN |
9781498796729 |
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1498796729 |
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