UID:
almafu_9958104587402883
Umfang:
1 online resource (417 p.)
ISBN:
1-281-74233-3
,
9786611742331
,
0-08-087435-5
Serie:
Pure and applied mathematics ; 116
Inhalt:
Differential manifolds and theoretical physics
Anmerkung:
Description based upon print version of record.
,
Front Cover; Differential Manifolds and Theoretical Physics; Copyright Page; Contents; Preface; Chapter 1. Introduction; Mathematical Models for Physical Systems; Chapter 2. Classical Mechanics; Mechanics of Many-Particle Systems; Lagrangian and Hamiltonian Formulation; Mechanical System with Constraints; Exercises; Chapter 3. Introduction to Differential Manifolds; Differential Calculus in Several Variables; The Concept of a Differential Manifold; Submanifolds; Tangent Vectors; Smooth Maps of Manifolds; Differentials of Functions; Exercises; Chapter 4. Differential Equations on Manifolds
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Vector Fields and Integral CurvesLocal Existence And Uniqueness Theory; The Global Flow of a Vector Field; Complete Vector Fields; Exercises; Chapter 5. The Tangent and Cotangent Bundles; The Topology and Manifold Structure of the Tangent Bundle; The Cotangent Space and the Cotangent Bundle; The Canonical 1-Form on T*X; Exercises; Chapter 6. Covariant 2-Tensors and Metric Structures; Covariant Tensors of Degree 2; The Index of a Metric; Riemannian and Lorentzian Metrics; Behavior Under Mappings; Induced Metrics on Submanifolds; Raising and Lowering Indices; The Gradient of a Function
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Partitions of UnityExistence of Metrics on a Differential Manifold; Topology and Critical Points of a Fuction; Exercises; Chapter 7. Lagrangian and Hamiltonian Mechanics for Holonomic Systems; Introduction; The Total Force Mapping; Forces of Constraint; Lagrange's Equations; Conservative Forces; The Legendre Transformation; Conservation of Energy; Hamilton's Equations; 2-Forms; Exterior Derivative; Canonical 2-Form on T*X; The Mappings # and b; Hamiltonian and Lagrangian Vector Fields; Time-Dependent Systems; Exercises; Chapter 8. Tensors; Tensors on a Vector Space; Tensor Fields on Manifolds
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The Lie DerivativeThe Bracket of Vector Fields; Vector Fields as Differential Operators; Exercises; Chapter 9. Differential Forms; Exterior Forms on a Vector Space; Orientation of Vector Spaces; Volume Element of a Metric; Differential Forms on a Manifold; Orientation of Manifolds; Orientation of Hypersurfaces; Interior Product; Exterior Derivative; Poincaré Lemma; De Rham Cohomology Groups; Manifolds with Boundary; Induced Orientation; Hodge *-Duality; Divergence and Laplacian Operators; Calculations in Three-Dimensional Euclidean Space; Calculations in Minkowski Spacetime
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Geometrical Aspects of Differential FormsSmooth Vector Bundles; Vector Subbundles; Kernel of a Differential Form; Integrable Subbundles and the Frobenius Theorem; Integral Manifolds; Maximal Integral Manifolds; Inaccessibility Theorem; Nonintegrable Subbundles; Vector-Valued Differential Forms; Exercises; Chapter 10. Integration of Differential Forms; The Integeral of a Differential Form; Strokes's Theorem; Transformation Properties of Interals; ?-Divergence of a Vector Field; Other Versions of Stroke's Theorem; Integration of Functions; The Classical Integral Theorems; Exercises
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Chapter 11. The Special Theory of Relativity
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English
Weitere Ausg.:
ISBN 0-12-200230-X
Sprache:
Englisch
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