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全新数学物理的几何方法()(英)舒茨9787510004513
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1 Some basic mathematics1.1 The space Rn and its topology1.2 Mappings1.3 Real analysis1.4 Group theory1.5 Linear algebra1.6 The algebra of square matrices1.7 Bibliography2 Dffferentiable manifolds and tensors2.1 Defmition of a manifold2.2 The sphere as a manifold. Other examples of manifolds2.4 Global considerations2.5 Curves2.6 Functions on M2.7 Vectors and vector fields2.8 Basis vectors and basis vector fields2.9 Fiber bundles2.10 Examples of fiber bundles2.11 A deeper look at fiber bundles2.12 Vector fields and integral curves2.13 Exponentiation of the operator d/dZ2.14 Lie brackesndnncoordinate bases2.15 When is a basis a coordinate basis?2.16 One-forms2.17 Examples of one-forms2.18 The Dirac delta function2.19 The gradient and the pictorial representation of a one-form2.20 Basis one-forms and components of one-forms2.21 Index notation2.22 Tensors and tensor fields2. Examples of tensors2.24 Components of tensors and the outer product2.25 Contraction2.26 Basis transformations2.27 Tensor oraios on components2.28 Functions and scalars2.29 The metric tensor on a vector space.0 The metric tensor field on a manifold.1 Spe relativity.2 Bibliography3 Lie derivatives and Lie groups3.1 Introduction: how a vector field maps a manifold into itself3.2 Lie dragging a function3.3 Lie dragging a vector field3.4 Lie derivatives3.5 Lie derivative of a one-form3.6 Submanifolds3.7 Frobenius theorem (vector field version)3.8 Proof of Frobenius theorem3.9 An example: the generators ors23.10 Invariance3.11 Killing vector fields3.12 Killing vectors and conserved quantities in particle dynamics3.13 Axial symmetry3.14 Abstract Lie groups3.15 Examples of Lie groups3.16 Lie algebras and their groups3.17 Realizations and representatidns3.18 Spherical symmetry, spherical harmonics and representations of the rotation group3.19 Bibliography4 Differential forms A The algebra and integral calculus of forms4.1 Definition of volume - the geometrical role of differential forms4.2 Notation and definitions for antisymmetric tensors4.3 Differential forms4.4 Manipulating differential forms4.5 Restriction of forms4.6 Fields of forms4.7 Handedness and orientability4.8 Volumes and integration on oriented manifolds4.9 N-vectors, duals, and the symbol4.10 Tensor densities4.11 Generalized Kronecker deltas4.12 Determinants and4.13 Metric volume elements B The differential calculus of forms and its applications4.14 The exterior derivative4.15 Notation for derivatives4.16 Familiar examples of exterior differentiation4.17 Integrability conditions for partial differential equations4.18 Exact forms4.19 Proof of the local exactness of closed forms4.20 Lie derivatives of forms4.21 Lie derivatives and exterior derivatives commute4.22 Stokes theorem4. Gauss theorem and the definition of divergence4.24 A glance at cohomology theory4.25 Differential forms and differential equations4.26 Frobenins theorem (differential forms version)4.27 Proof of the equivalence of the two versions of Frobenius theorem4.28 Conservation laws4.29 Vector spherical harmonics4.30 Bibliography5 Applications in physics A Thermodynamics5.1 Simple systems5.2 Maxwell and other mathematical identities5.3 Coite thermodynamic systems: Caratheodorys theorem B Hamilton/an mechanics5.4 Hamiltodian vector fields5.5 Canonical transformations5.6 Map between vectors and one-forms provided by5.7 Poisson bracket5.8 Many-particle systems: symplectic forms5.9 Linear dynamical systems: the symplectic inner producndcnserved quantities5.10 Fiber bundle structure of the Hamiltonian equations C Electromagnetism5.11Rewriting Maxwells equations using differential forms5.12 Charge and topology5.13 The vector potential5.14 Plane waves: a simple example D Dynamics of a perfect fluid5.15 Role of Lie derivatives5.16 The comoving time-derivative5.17 Equation of motion5.18 Conservation of vorticityE Cosmology5.19 The cosmological principle5.20 Lie algebra of maximal symmetry5.21 The metric of a spherically symmetric three-space5.22 Construction of the six Killing vectors5. Open, closed, and flat universes5.24 Bibliography6 Connections for Riemnnnian manifolds and gauge theories6.1 Introduction6.2 Parallelism on curved surfaces6.3 The covariant derivative6.4 Components: covariant derivatives of the basis6.5 Torsion6.6 Geodesics6.7 Normal coordinates6.8 Riemann tensor6.9 Geometric interpretation of the Riemann tensor6.10 Flat spaces6.11 Compatibility of the connection with volume-measure or the metric6.12 Metric connections6.13 The affine connection and the equivalence principle6.14 Connections and gauge theories: the example of electromagnetism6.15 BibfiographyAppendix: solutions and hints for selected exercisesNotationIndex
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