Differential manifolds and theoretical physics / W.D. Curtis, F.R. Miller.
Series Pure and applied mathematics (Academic Press): 116.Editor: Orlando [Fla.] : Academic Press, 1985Descripción: xix, 394 p. : ill. ; 24 cmISBN: 012200230X (alk. paper)Tema(s): Geometry, Differential | Mechanics | Field theory (Physics) | Differentiable manifoldsOtra clasificación: 58-01 | 53-01Chapter 1. Introduction Mathematical Models for Physical Systems [1] Chapter 2. Classical Mechanics Mechanics of Many-Particle Systems [5] Lagrangian and Hamiltonian Formulation [7] Mechanical Systems with Constraints [11] Exercises [14] Chapter 3. Introduction to Differential Manifolds Differential Calculus in Several Variables [16] The Concept of a Differential Manifold [23] Submanifolds [26] Tangent Vectors [28] Smooth Maps of Manifolds [33] Differentials of Functions [35] Exercises [36] Chapter 4. Differential Equations on Manifolds Vector Fields and Integral Curves [39] Local Existence and Uniqueness Theory [40] The Global Flow of a Vector Field [53] Complete Vector Fields [55] Exercises [57] Chapter 5. The Tangent and Cotangent Bundles The Topology and Manifold Structure of the Tangent Bundle [61] The Cotangent Space and the Cotangent Bundle [66] The Canonical 1-Form on T*X [68] Exercises [70] Chapter 6. Covariant 2-Tensors and Metric Structures Covariant Tensors of Degree 2 [72] The Index of a Metric [74] Riemannian and Lorentzian Metrics [74] Behavior Under Mappings [77] Induced Metrics on Submanifolds [79] Raising and Lowering Indices [83] The Gradient of a Function [84] Partitions of Unity [84] Existence of Metrics on a Differential Manifold [87] Topology and Critical Points of a Function [90] Exercises [92] Chapter 7.Lagrangian and Hamiltonian Mechanics for Holonomic Systems Introduction [94] The Total Force Mapping [95] Forces of Constraint [96] Lagrange’s Equations [99] Conservative Forces [99] The Legendre Transformation [103] Conservation of Energy [105] Hamilton’s Equations [106] 2-Forms [110] Exterior Derivative [112] Canonical 2-Form on T*X [114] The Mappings # and b [114] Hamiltonian and Lagrangian Vector Fields [115] Time-Dependent Systems [121] Exercises [124] Chapter 8.Tensors Tensors on a Vector Space [127] Tensor Fields on Manifolds [129] The Lie Derivative [132] The Bracket of Vector Fields [135] Vector Fields as Differential Operators [137] Exercises [138] Chapter 9. Differential Forms Exterior Forms on a Vector Space [141] Orientation of Vector Spaces [146] Volume Element of a Metric [149] Differential Forms on a Manifold [150] Orientation of Manifolds [151] Orientation of Hypersurfaces [154] Interior Product [156] Exterior Derivative [156] Poincard Lemma [161] De Rham Cohomology Groups [161] Manifolds with Boundary [162] Induced Orientation [163] Hodge *-Duality [165] Divergence and Laplacian Operators [168] Calculations in Three-Dimensional Euclidean Space [168] Calculations in Minkowski Spacetime [170] Geometrical Aspects of Differential Forms [171] Smooth Vector Bundles [172] Vector Subbundles [172] Kernel of a Differential Form [173] Integrable Subbundles and the Frobenius Theorem [176] Integral Manifolds [184] Maximal Integral Manifolds [185] Inaccessibility Theorem [187] Nonintegrable Subbundles [188] Vector-Valued Differential Forms [189] Exercises [191] Chapter 10. Integration of Differential Forms The Integral of a Differential Form [196] Stokes’s Theorem [199] Transformation Properties of Integrals [201] w-Divergence of a Vector Field [203] Other Versions of Stokes’s Theorem [204] Integration of Functions [207] The Classical Integral Theorems [208] Exercises [210] The Special Theory of Relativity Basic Concepts and Relativity Groups [213] Relativistic Law of Velocity Addition [220] Relativity of Simultaneity [222] Relativistic Length Contraction [222] Relativistic Time Dilation [223] The Invariant Spacetime Interval [223] The Proper Lorentz Group and the Poincaré Group [224] The Spacetime Manifold of Special Relativity [225] Relativistic Time Units [227] Accelerated Motion—A Space Odyssey [229] Energy and Momentum [233] Relativistic Correction to Newtonian Mechanics [234] Conservation of Energy and Momentum [235] Mass and Energy [236] Changes in Rest Mass [236] Summary [237] Exercises [237] Electromagnetic Theory The Lorentz Force Law and the Faraday Tensor [239] The 4-Current [243] Doppler Effect [245] Maxwell’s Equations [246] The Electromagnetic Plane Wave [248] The 4-Potential [250] Existence of Scalar and Vector Potentials in R3 [251] Exercises [253] The Mechanics of Rigid Body Motion Hamiltonian Systems and Equivalent Models [255] The Rigid Body [256] 0(3) and SO(3) [256] Space and Body Representations [259] The Geometry of Rigid Body Motion [261] Left-Invariant 1-Form [263] Symmetry Group [264] Adjoint Representation [264] Momentum Mapping [265] Coadjoint Representation [266] Space Motions with Specified Momentum [266] Coadjoint Orbits and Body Motions [267] Special Properties of 50(3) [271] Stationary Rotations [274] Classical Interpretation—Inertial Tensor, Principal Axes 274 Stability of Stationary Rotations [277] Poinsot Construction [280] Euler Equations [282] Phase Plane Analysis of Stability [283] Exercises [284] Chapter 14. Lie Groups Lie Groups and Their Lie Algebras [286] Exponential Mapping [289] Canonical Coordinates [289] Subgroups and Homomorphisms [290] Adjoint Representation [291] Invariant Forms [292] Coset Spaces and Actions [293] Exercises [296] Chapter 15. Geometrical Models Geometrical Mechanical Systems [297] Liouville’s Theorem [298] Variational Principles [300] Forces [301] Fixed Energy Systems [304] Configuration Projections [305] Lorentz Force Law [306] Pseudomechanical Systems [306] Restriction Mappings [307] Rigid Body and Torque [308] Gauge Group Actions [310] Moving Frames and Goedesic Motion [311] Basic Theorem Local (Lemma 15.36) [314] Basic Theorem Global (Theorem 15.39) [316] Principal Bundle Model Using a Special Frame [319] The Souriau Equations [321] Structure of the Lie Algebra of the Lorentz Group [322] Construction of a Gauge Invariant 2-Form [322] Curvature Form [327] The Souriau GMS [328] Appendix: Conservation Laws [329] Exercises [332] Principal Bundles and Connections; Gauge Fields and Classical Particles Principal Bundles [335] Reduction of the Structural Group [337] Connections on Principal Bundles [337] Horizontal Lifts of Vectors [338] Curvature Form and Integrability Theorem [339] Horizontal Lifts of Curves [341] Associated Bundles [341] Parallel Transport [342] Gauge Fields and Classical Particles [342] Natural 2-Form on Coadjoint Orbits [343] Pseudomechanical System for Particles in a Gauge Field 345 Sternberg’s Theorem [346] Geometrical-Mechanical System for Particles in a Gauge Field [347] Affine Group Model [349] Exercises [351] Quantum Effects, Line Bundles, and Holonomy Groups Quantum Effects [354] Probability Amplitudes [355] Probability Amplitude Phase Factors [355] DeBroglie and Feynman [356] Phase Factors and 1-Forms [356] COW and Bohm-Aharanov Experiments [358] Complex Line Bundles and Holonomy Groups [360] Integral Condition for Curvature Form [362] Bundle Description of Phase Factor Calculation [365] Remarks—Geometric Quantization [366] Holonomy and Curvature for General Lie Groups [367] Exercises [368] Physical Laws for the Gauge Fields Gauss’s Law in Electromagnetic Theory [371] Charge Conservation [372] Curvature and Bundle-Valued Differential Forms [373] Covariant Exterior Derivative [375] Covariant Derivative of Sections and Parallel Transport 376 The Group of Gauge Transformations [377] The Killing form [379] The Source Equation and Currents for Gauge Fields [379] Exercises [382] Bibliography [387] Index [389]
| Item type | Home library | Shelving location | Call number | Materials specified | Status | Date due | Barcode |
|---|---|---|---|---|---|---|---|
Libros
|
Instituto de Matemática, CONICET-UNS | Libros ordenados por tema | 58 C978 (Browse shelf) | Available | A-9308 |
Browsing Instituto de Matemática, CONICET-UNS shelves, Shelving location: Libros ordenados por tema Close shelf browser
| 58 C7525 Noncommutative geometry, quantum fields and motives / | 58 C889 Applicable differential geometry / | 58 C889 Applicable differential geometry / | 58 C978 Differential manifolds and theoretical physics / | 58 D315 Deformation theory and symplectic geometry : | 58 D522 Supermanifolds / | 58 D567 Éléments d'analyse / |
Includes index.
Bibliografía: p. 387-388.
Chapter 1. Introduction --
Mathematical Models for Physical Systems [1] --
Chapter 2. Classical Mechanics --
Mechanics of Many-Particle Systems [5] --
Lagrangian and Hamiltonian Formulation [7] --
Mechanical Systems with Constraints [11] --
Exercises [14] --
Chapter 3. Introduction to Differential Manifolds --
Differential Calculus in Several Variables [16] --
The Concept of a Differential Manifold [23] --
Submanifolds [26] --
Tangent Vectors [28] --
Smooth Maps of Manifolds [33] --
Differentials of Functions [35] --
Exercises [36] --
Chapter 4. Differential Equations on Manifolds --
Vector Fields and Integral Curves [39] --
Local Existence and Uniqueness Theory [40] --
The Global Flow of a Vector Field [53] --
Complete Vector Fields [55] --
Exercises [57] --
Chapter 5. The Tangent and Cotangent Bundles --
The Topology and Manifold Structure of the Tangent Bundle [61] --
The Cotangent Space and the Cotangent Bundle [66] --
The Canonical 1-Form on T*X [68] --
Exercises [70] --
Chapter 6. Covariant 2-Tensors and Metric Structures --
Covariant Tensors of Degree 2 [72] --
The Index of a Metric [74] --
Riemannian and Lorentzian Metrics [74] --
Behavior Under Mappings [77] --
Induced Metrics on Submanifolds [79] --
Raising and Lowering Indices [83] --
The Gradient of a Function [84] --
Partitions of Unity [84] --
Existence of Metrics on a Differential Manifold [87] --
Topology and Critical Points of a Function [90] --
Exercises [92] --
Chapter 7.Lagrangian and Hamiltonian Mechanics for Holonomic Systems --
Introduction [94] --
The Total Force Mapping [95] --
Forces of Constraint [96] --
Lagrange’s Equations [99] --
Conservative Forces [99] --
The Legendre Transformation [103] --
Conservation of Energy [105] --
Hamilton’s Equations [106] --
2-Forms [110] --
Exterior Derivative [112] --
Canonical 2-Form on T*X [114] --
The Mappings # and b [114] --
Hamiltonian and Lagrangian Vector Fields [115] --
Time-Dependent Systems [121] --
Exercises [124] --
Chapter 8.Tensors --
Tensors on a Vector Space [127] --
Tensor Fields on Manifolds [129] --
The Lie Derivative [132] --
The Bracket of Vector Fields [135] --
Vector Fields as Differential Operators [137] --
Exercises [138] --
Chapter 9. Differential Forms --
Exterior Forms on a Vector Space [141] --
Orientation of Vector Spaces [146] --
Volume Element of a Metric [149] --
Differential Forms on a Manifold [150] --
Orientation of Manifolds [151] --
Orientation of Hypersurfaces [154] --
Interior Product [156] --
Exterior Derivative [156] --
Poincard Lemma [161] --
De Rham Cohomology Groups [161] --
Manifolds with Boundary [162] --
Induced Orientation [163] --
Hodge *-Duality [165] --
Divergence and Laplacian Operators [168] --
Calculations in Three-Dimensional Euclidean Space [168] --
Calculations in Minkowski Spacetime [170] --
Geometrical Aspects of Differential Forms [171] --
Smooth Vector Bundles [172] --
Vector Subbundles [172] --
Kernel of a Differential Form [173] --
Integrable Subbundles and the Frobenius Theorem [176] --
Integral Manifolds [184] --
Maximal Integral Manifolds [185] --
Inaccessibility Theorem [187] --
Nonintegrable Subbundles [188] --
Vector-Valued Differential Forms [189] --
Exercises [191] --
Chapter 10. Integration of Differential Forms --
The Integral of a Differential Form [196] --
Stokes’s Theorem [199] --
Transformation Properties of Integrals [201] --
w-Divergence of a Vector Field [203] --
Other Versions of Stokes’s Theorem [204] --
Integration of Functions [207] --
The Classical Integral Theorems [208] --
Exercises [210] --
The Special Theory of Relativity --
Basic Concepts and Relativity Groups [213] --
Relativistic Law of Velocity Addition [220] --
Relativity of Simultaneity [222] --
Relativistic Length Contraction [222] --
Relativistic Time Dilation [223] --
The Invariant Spacetime Interval [223] --
The Proper Lorentz Group and the Poincaré Group [224] --
The Spacetime Manifold of Special Relativity [225] --
Relativistic Time Units [227] --
Accelerated Motion—A Space Odyssey [229] --
Energy and Momentum [233] --
Relativistic Correction to Newtonian Mechanics [234] --
Conservation of Energy and Momentum [235] --
Mass and Energy [236] --
Changes in Rest Mass [236] --
Summary [237] --
Exercises [237] --
Electromagnetic Theory --
The Lorentz Force Law and the Faraday Tensor [239] --
The 4-Current [243] --
Doppler Effect [245] --
Maxwell’s Equations [246] --
The Electromagnetic Plane Wave [248] --
The 4-Potential [250] --
Existence of Scalar and Vector Potentials in R3 [251] --
Exercises [253] --
The Mechanics of Rigid Body Motion --
Hamiltonian Systems and Equivalent Models [255] --
The Rigid Body [256] --
0(3) and SO(3) [256] --
Space and Body Representations [259] --
The Geometry of Rigid Body Motion [261] --
Left-Invariant 1-Form [263] --
Symmetry Group [264] --
Adjoint Representation [264] --
Momentum Mapping [265] --
Coadjoint Representation [266] --
Space Motions with Specified Momentum [266] --
Coadjoint Orbits and Body Motions [267] --
Special Properties of 50(3) [271] --
Stationary Rotations [274] --
Classical Interpretation—Inertial Tensor, Principal Axes 274 Stability of Stationary Rotations [277] --
Poinsot Construction [280] --
Euler Equations [282] --
Phase Plane Analysis of Stability [283] --
Exercises [284] --
Chapter 14. Lie Groups --
Lie Groups and Their Lie Algebras [286] --
Exponential Mapping [289] --
Canonical Coordinates [289] --
Subgroups and Homomorphisms [290] --
Adjoint Representation [291] --
Invariant Forms [292] --
Coset Spaces and Actions [293] --
Exercises [296] --
Chapter 15. Geometrical Models --
Geometrical Mechanical Systems [297] --
Liouville’s Theorem [298] --
Variational Principles [300] --
Forces [301] --
Fixed Energy Systems [304] --
Configuration Projections [305] --
Lorentz Force Law [306] --
Pseudomechanical Systems [306] --
Restriction Mappings [307] --
Rigid Body and Torque [308] --
Gauge Group Actions [310] --
Moving Frames and Goedesic Motion [311] --
Basic Theorem Local (Lemma 15.36) [314] --
Basic Theorem Global (Theorem 15.39) [316] --
Principal Bundle Model Using a Special Frame [319] --
The Souriau Equations [321] --
Structure of the Lie Algebra of the Lorentz Group [322] --
Construction of a Gauge Invariant 2-Form [322] --
Curvature Form [327] --
The Souriau GMS [328] --
Appendix: Conservation Laws [329] --
Exercises [332] --
Principal Bundles and Connections; Gauge Fields and Classical Particles --
Principal Bundles [335] --
Reduction of the Structural Group [337] --
Connections on Principal Bundles [337] --
Horizontal Lifts of Vectors [338] --
Curvature Form and Integrability Theorem [339] --
Horizontal Lifts of Curves [341] --
Associated Bundles [341] --
Parallel Transport [342] --
Gauge Fields and Classical Particles [342] --
Natural 2-Form on Coadjoint Orbits [343] --
Pseudomechanical System for Particles in a Gauge Field 345 Sternberg’s Theorem [346] --
Geometrical-Mechanical System for Particles in a Gauge Field [347] --
Affine Group Model [349] --
Exercises [351] --
Quantum Effects, Line Bundles, and Holonomy Groups --
Quantum Effects [354] --
Probability Amplitudes [355] --
Probability Amplitude Phase Factors [355] --
DeBroglie and Feynman [356] --
Phase Factors and 1-Forms [356] --
COW and Bohm-Aharanov Experiments [358] --
Complex Line Bundles and Holonomy Groups [360] --
Integral Condition for Curvature Form [362] --
Bundle Description of Phase Factor Calculation [365] --
Remarks—Geometric Quantization [366] --
Holonomy and Curvature for General Lie Groups [367] --
Exercises [368] --
Physical Laws for the Gauge Fields --
Gauss’s Law in Electromagnetic Theory [371] --
Charge Conservation [372] --
Curvature and Bundle-Valued Differential Forms [373] --
Covariant Exterior Derivative [375] --
Covariant Derivative of Sections and Parallel Transport 376 The Group of Gauge Transformations [377] --
The Killing form [379] --
The Source Equation and Currents for Gauge Fields [379] --
Exercises [382] --
Bibliography [387] --
Index [389] --
MR, REVIEW #
Click on an image to view it in the image viewer
There are no comments on this title.