Analysis, manifolds and physics

Автор(ы):Choquet-Bruhat Y., Dewitt-Morette C.,Dillard-Bleick M.
06.10.2007
Год изд.:1970
Издание:2
Описание: Can this book, now polished by usage, serve as a text for an advanced physical mathematics course? This question raises another question: What is the function of a text book for graduate studies? In our times of rapidly expanding knowledge, a teacher looks for a book which will provide a broader base for future developments than can be covered in one or two semesters of lectures and a student hopes that his purchase will serve him for many years. A reference book which can be used as a text is an answer to their needs. This is what this book is intended to be.
Оглавление:
Analysis, manifolds and physics — обложка книги.
I. Review of Fundamental Notions of Analysis [1]
A. Set Theory, Definitions [1]
  1.Sets [1]
  2. Mappings [2]
  3. Relations [5]
  4. Orderings [5]
B. Algebraic Structures, Definitions [6]
  1. Groups [7]
  2. Rings [8]
  3. Modules [8]
  4. Algebras [9]
  5. Linear spaces [9]
C. Topology [11]
  1. Definitions [11]
  2. Separation [13]
  3.Вазе [14]
  4. Convergence [14]
  5. Covering and compactness [15]
  6. Connectedness [16]
  7. Continuous mappings [17]
  8. Multiple connectedness [19]
  9. Associated topologies [20]
  10. Topology related to other structures [21]
  11. Metric spaces [23]
    metric spaces [23]
    Cauchy sequence; completeness [25]
  12. Banach spaces [26]
    normed vector spaces [27]
    Banach spaces [28]
    strong and weak topology; compactedness [29]
  13. Hubert spaces [30]
D. Integration [31]
  1. Introduction [32]
  2. Measures [33]
  3. Measure spaces [34]
  4. Measurable functions [40]
  5. Integrable functions [41]
  6. Integration on locally compact spaces [46]
  7. Signed and complex measures [49]
  8. Integration of vector valued functions [50]
  9. (?) space [52]
  10. If space [53]
E. Key Theorems in Linear Functional Analysis [57]
  1. Bounded linear operators [57]
  2. Compact operators [61]
  3. Open mapping and closed graph theorems [63]
Problems and Exercises [64]
  Problem 1: Clifford algebra; Spin(4) [64]
  Exercise 2: Product topology [68]
  Problem 3: Strong and weak topologies in (?) [69]
  Exercise 4: Holder spaces [70]
  See Problem VI 4: Application to the Schrodinger equation [70]
II. Differential Calculus on Banach Spaces [71]
A. Foundations [71]
  1. Definitions. Taylor expansion [71]
  2. Theorems [73]
  3. Diffeomorphisms [74]
  4. The Euler equation [76]
  5. The mean value theorem [78]
  6. Higher order differentials [79]
B. Calculus of Variations [82]
  1. Necessary conditions for minima [82]
  2. Sufficient conditions [83]
  3. Lagrangian problems [86]
C. Implicit Function Theorem. Inverse Function Theorem [88]
  1. Contracting mapping theorems [88]
  2. Inverse function theorem [90]
  3. Implicit function theorem [91]
  4. Global theorems [92]
D. Differential Equations [94]
  1. First order differential equation [94]
  2. Existence and uniqueness theorems for the lipschitzian case [95]
Problems and Exercises [98]
  Problem 1: Banach spaces, first variation, linearized equation [98]
  Problem 2: Taylor expansion of the action; Jacobi fields; the Feynman [100]
  Green function; the Van Vleck matrix; conjugate points; caustics
  Problem 3: Euler-Lagrange equation; the small disturbance equation; the soap bubble problem; Jacobi fields [105]
III. Differentiate Manifolds, Finite Dimensional Case [111]
A. Definitions [111]
  1. Differentiable manifolds [111]
  2. Diffeomorphisms [115]
  3. Lie groups [116]
B. Vector Fields; Tensor Fields [117]
  1. Tangent vector space at a point [117]
    tangent vector as a derivation [118]
    tangent vector defined by transformation properties [120]
    tangent vector as an equivalence class of curves [121]
    images under differentiable mappings [121]
  2. Fibre bundles [124]
    definition [125]
    bundle morphisms [127]
    tangent bundle [127]
    frame bundle [128]
    principal fibre bundle [129]
  3. Vector fields [132]
    vector fields [132]
    moving frames [134]
    images under diffeomorphisms [134]
  4. Covariant vectors; cotangent bundles [135]
    dual of the tangent space [135]
    space of differentials [137]
    cotangent bundle [138]
    reciprocal images [138]
  5. Tensors at a point [138]
    tensors at a point [138]
    tensor algebra [140]
  6. Tensor bundles; tensor fields [142]
C. Groups of Transformations [143]
  1. Vector fields as generators of transformation groups [143]
  2. Lie derivatives [147]
  3. Invariant tensor fields [150]
D. Lie Groups [152]
  1. Definitions; notations [152]
  2. Left and right translations; Lie algebra; structure constants [155]
  3. One-parameter subgroups [158]
  4. Exponential mapping; Taylor expansion; canonical coordinates [160]
  5. Lie groups of transformations; realization [162]
  6. Adjoint representation [166]
  7. Canonical form, Maurer-Cartan form [168]
Problems and Exercises [169]
  Problem 1: Change of coordinates on a fiber bundle, configuration space, phase space [169]
  Problem 2: Lie algebras of Lie groups [172]
  Problem 3: The strain tensor [177]
  Problem 4: Exponential map; Taylor expansion; adjoint map; left and right differentials; Haar measure [178]
  Problem 5: The group manifolds of SO(3) and SU(2) [181]
  Problem 6: The 2-sphere [190]
IV. Integration on Manifolds [195]
A. Exterior Differential Forms [195]
  1. Exterior algebra [195] exterior product [196]
    local coordinates; strict components [197]
    change of basis [199]
  2. Exterior differentiation [200]
  3. Reciprocal image of a form (pull back) [203]
  4. Derivations and antiderivations [205]
    definitions [206]
    interior product [207]
  5. Forms defined on a Lie group [208]
    invariant forms [208]
    Maurer-Cartan structure equations [208]
  6. Vector valued differential forms [210]
B. Integration [212]
  1. Integration [212]
    orientation [212]
    odd forms [212]
    integration of n-forms in (?) [213]
    partitions of unify [214]
    properties of integrals [215]
  2. Stokes' theorem [216]
    p-chains [217]
    integrals ofp-forms on p-chains [217]
    boundaries [218]
    mappings of chains [219]
    proof of Stokes' theorem [221]
  3. Global properties [222]
    homology and cohomology [222]
    0-forms and 0-chains [223]
    Betti numbers [224]
    Poincare lemmas [224]
    de Rham and Poincare duality theorems [226]
C. Exterior Differential Systems [229]
  1. Exterior equations [229]
  2. Single exterior equation [229]
  3. Systems of exterior equations [232]
    ideal generated by a system of exterior equations [232]
    algebraic equivalence [232]
    solutions [233]
    examples [235]
  4. Exterior differential equations [236]
    integral manifolds [236]
    associated Pfaff systems [237]
    generic points [238]
    closure [238]
  5. Mappings of manifolds [239]
    introduction [239]
    immersion [241]
    embedding [241]
    submersion [242]
  6. Pfaff systems [242]
    complete integrability [243]
    Frobenius theorem [243]
    integrability criterion [245]
    examples [246]
    dual form of the Frobenius theorem [248]
  7. Characteristic system [250]
    characteristic manifold [250]
    example: first order partial differential equations [250]
    complete integrability [253]
    construction of integral manifolds [254]
    Cauchy problem [256]
    examples [259]
  8. Invariants [261]
    invariant with respect to a Pfaff system [261]
    integral invariants [263]
  9. Example: Integral invariants of classical dynamics [265]
    Liouville theorem [266]
    canonical transformations [267]
  10. Symplectic structures and hamiltonian systems [267]
Problems and Exercises [270]
  Problem 1: Compound matrices [270]
  Problem 2: Poincare lemma. Maxwell equations, wormholes [271]
  Problem 3: Integral manifolds [271]
  Problem 4: First order partial differential equations, Hamilton-Jacobi equations, lagrangian manifolds [272]
  Problem 5: First order partial differential equations, catastrophes [277]
  Problem 6: Darboux theorem [281]
  Problem 7: Time dependent hamiltonians [283]
  See Problem VI 1 1 paragraph c: Electromagnetic shock waves
V. Riemannian Manifolds. Kahlerian Manifolds [285]
A. The Riemannian Structure [285]
  1. Preliminaries [285]
    metric tensor [285]
    hyperbolic manifold [287]
  2. Geometry of submanifolds, induced metric [290]
  3. Existence of a riemannian structure [292]
    proper structure [292]
    hyperbolic structure [293]
    Euler-Poincare characteristic [293]
  4. Volume element. The star operator [294]
    volume element [294]
    star operator [295]
  5. Isometries [298]
B. Linear Connections [300]
  1. Linear connections [300]
    covariant derivative [301]
    connection forms [301]
    parallel translation [302]
    affine geodesic [302]
    torsion and curvature [305]
  2. Riemannian connection [308]
    definitions [309]
    locally flat manifolds [310]
  3. Second fundamental form [312]
  4. Differential operators [316]
    exterior derivative [316]
    operator S [317]
    divergence [317]
    laplacian [318]
C. Geodesies [320]
  1. Arc length [320]
  2. Variations [321]
    Euler equations [323]
    energy integral [324]
  3. Exponential mapping [325]
    definition [325]
    normal coordinates [326]
  4. Geodesies on a proper riemannian manifold [327]
    properties [327]
    geodesic completeness [330]
  5. Geodesies on a hyperbolic manifold [330]
D. Almost Complex andKdhlerian Manifolds [330]
Problems and Exercises [336]
  Problem 1: Maxwell equation; gravitational radiation [336]
  Problem 2: The Schwarzschild solution [341]
  Problem 3: Geodetic motion; equation of geodetic deviation; exponentiation; conjugate points [344]
  Problem 4: Causal structures; conformal spaces; Weyl tensor [350]
Vbis. Connections on a Principal Fibre Bundle [357]
A. Connections on a Principal Fibre Bundle [357]
  1. Definitions [357]
  2. Local connection 1-forms on the base manifold [362]
    existence theorems [362]
    section canonically associated with a trivialization [363]
    potentials [364]
    change of trivialization [364]
    examples [366]
  3. Covariant derivative [367] associated bundles [367]
    parallel transport [369]
    covariant derivative [370]
    examples [371]
  4. Curvature [372]
    definitions [372]
    Cartan structural equation [373]
    local curvature on the base manifold [374]
    field strength [375]
    Bianchi identities [375]
  5. Linear connections [376]
    definition [376]
    soldering form, torsion form [376]
    torsion structural equation [376]
    standard horizontal (basic) vector field [378]
    curvature and torsion on the base manifold [378]
    bundle homomorphism [380]
    metric connection [381]
B. Holonomy [381]
  1. Reduction [381]
  2. Holonomy groups [386]
C. Characteristic Classes and Invariant Curvature Integrals [390]
  1. Characteristic classes [390]
  2. Gauss-Bonnet theorem and Chern numbers [395]
  3. The Atiyah-Singer index theorem [396]
Problems and Exercises [401]
  Problem 1 : The geometry of gauge fields [401]
  Problem 2: Charge quantization. Monopoles [408]
  Problem 3: Instanton solution of euclidean SU(2) Yang-Mills theory (connection on a non-trivial SU(2) bundle over S4) [411]
  Problem 4: Spin structure; spinors; spin connections [415]
VI. Distributions [423]
A. Test Functions [423]
  1. Seminorms [423]
    definitions [423]
    Hahn-Banach theorem [424]
    topology defined by a family of seminorms [424]
  2. (?) -spaces [427]
    definitions [427]
    inductive limit topology [429]
    convergence in (?) and (?) [430]
    examples of functions in (?) [431]
    truncating sequences [434]
    density theorem [434]
    Distributions [435]
  1. Definitions [435]
    distributions [435]
    measures; Dirac measures and Leray forms [437]
    distribution of order p [439]
    support of a distribution [441]
    distributions with compact support [441]
  2. Operations on distributions [444]
    sum [444]
    product by C(?) function [444]
    direct product [445]
    derivations [446]
    examples [447]
    inverse derivative [450]
  3. Topology on (?) [453]
    weak star topology [453]
    criterion of convergence [454]
  4. Change of variables in (?) [456]
    change of variables in (?) [456]
    transformation of a distribution under a diffeomorphism [457]
    invariance [459]
  5. Convolution [459]
    convolution algebra (?) [459]
    convolution algebra (?) and (?) [462]
    derivation and translation of a convolution product [464]
    regularization [465]
    support of a convolution [465]
    equations of convolution [466]
    differential equation with constant coefficients [469]
    systems of convolution equations [470]
    kernels [471]
  6. Fourier transform [474]
    Fourier transform ofintegrable functions [474]
    tempered distributions [476]
    Fourier transform of tempered distributions [476]
    Paley-Wiener theorem [477]
    Fourier transform of a convolution [478]
  7. Distribution on a (?) paracompact manifold [480]
  8. Tensor distributions [482]
C. Sobolev Spaces and Partial Differential Equations [486]
  1. Sobolev spaces [486] properties [487]
    density theorems [488]
    (?) spaces [489]
    Fourier transform [490]
    Plancherel theorem [490]
    Sobolev's inequalities [491]
  2. Partial differential equations [492]
    definitions [492]
    Cauchy-Kovalevski theorem [493]
    classifications [494]
  3. Elliptic equations; laplacians [495]
    elementary solution of Laplace's equation [495]
    subharmonic distributions [496]
    potentials [496]
    energy integral. Green's formula, unicity theorem [499]
    Liouville's theorem [500]
    boundary-value problems [502]
    Green function [503]
    introduction to hilbertian methods; generalized Dirichlet problem [505]
    hilbertian methods [507]
    example: Neumann problem [509]
  4. Parabolic equations[510]
    heat diffusion [510]
  5. Hyperbolic equation; wave equations [511]
    elementary solution of the wave equation [511]
    Cauchy problem [512]
    energy integral, unicity theorem [513]
    existence theorem [515]
  6. Leray theory of hyperbolic systems [516]
  7. Second order systems; propagators [522]
Problems and Exercises [525]
  Problem 1: Bounded distributions [525]
  Problem 2: Laplacian of a discontinuous function [527]
  Exercise 3: Regularized functions [528]
  Problem 4: Application to the Schrodinger equation [528]
  Exercise 5: Convolution and linear continuous responses [530]
  Problem 6: Fourier transforms of exp(?) and exp(?) [531]
  Problem 7: Fourier transforms of Heaviside functions and (?) [532]
  Problem 8: Dirac bitensors [533]
  Problem 9: Legendre condition [533]
  Problem 10: Hyperbolic equations; characteristics [534]
  Problem 11: Electromagnetic shock waves [535]
  Problem 12: Elementary solution of the wave equation [538]
  Problem 13: Elementary kernels of the harmonic oscillator [538]
VII. Differentiable Manifolds, Infinite Dimensional Case [543]
A. Infinite-Dimensional Manifolds [543]
  1. Definitions and general properties [543]
    E -manifolds [543]
    differentiable functions [544]
    tangent vector [544]
    vector and tensor field [545]
    differential of amapping [546]
    submanifold [547]
    immersion, embedding, submersion [549]
    flow of a vector field [551]
    differential forms [551]
  2. Symplectic structures and hamiltonian systems [552]
    definitions [552]
    complex structures [552]
    canonical symplectic form [554]
    symplectic transformation [554]
    hamiltonian vector field [554]
    conservation of energy theorem [555]
    riemannian manifolds [555]
B. Theory of Degree; Leray-Schauder Theory [556]
  1. Definition for finite dimensional manifolds [557]
    degree [557]
    integral formula for the degree of a function [558]
    continuous mappings [560]
  2. Properties and applications [561]
    fundamental theorem [561]
    Borsuk's theorem [562]
    Brouwer's fixed point theorem [562]
    product theorem [563]
  3. Leray-Schauder theory [563]
    definitions [563]
    compact mappings [564]
    degree of a compact mapping [564]
    Schauder fixed point theorem [565]
    Leray-Schauder theorem [565]
C. Morse Theory [567]
  1. Introduction [567]
  2. Definitions and theorems [567]
  3. Index of a critical point [571]
  4. Critical neck theorem [572]
D. Cylindrical Measures, Wiener Integral [573]
  1. Introduction [573]
  2. Promeasures and measures on a locally convex space [575]
    projective system [575]
    promeasures [576]
    image of apromeasure [578]
    integration with respect to apromeasure of a cylindrical function [578]
    Fourier transforms [579]
  3. Gaussian promeasures [581]
    gaussian measures on (?) [581]
    gaussian promeasures [582]
    gaussian promeasures on Hilbert spaces [583]
  4. The Wiener measure [583]
    Wiener integral [586]
    sequential Wiener integral [587]
Problems and Exercises [589]
  Problem A: The Klein-Gordon equation [589]
  Problem B: Application of the Leray-S chauder theorem [591]
  Problem C1 : The Reeb theorem [592]
  Problem C2: The method of stationary phase [593]
  Problem D1: A metric on the space of paths with fixed end points [596]
  Problem D2: Measures invariant under translation [597]
  Problem D3: Cylindrical ст-field of C([a,b]) [597]
  Problem D4: Generalized Wiener integral of a cylindrical function [598]
References [603]
Symbols [611]
Index [617]
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