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Geometrical Methods of Mathematical Physics

Book Description

In recent years the methods of modern differential geometry have become of considerable importance in theoretical physics and have found application in relativity and cosmology, high-energy physics and field theory, thermodynamics, fluid dynamics and mechanics. This textbook provides an introduction to these methods - in particular Lie derivatives, Lie groups and differential forms - and covers their extensive applications to theoretical physics. The reader is assumed to have some familiarity with advanced calculus, linear algebra and a little elementary operator theory. The advanced physics undergraduate should therefore find the presentation quite accessible. This account will prove valuable for those with backgrounds in physics and applied mathematics who desire an introduction to the subject. Having studied the book, the reader will be able to comprehend research papers that use this mathematics and follow more advanced pure-mathematical expositions.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Contents
  6. Preface
  7. 1. Some basic mathematics
    1. 1.1 The space Rn and its topology
    2. 1.2 Mappings
    3. 1.3 Real analysis
    4. 1.4 Group theory
    5. 1.5 Linear algebra
    6. 1.6 The algebra of square matrices
    7. 1.7 Bibliography
  8. 2. Differentiable manifolds and tensors
    1. 2.1 Definition of a manifold
    2. 2.2 The sphere as a manifold
    3. 2.3 Other examples of manifolds
    4. 2.4 Global considerations
    5. 2.5 Curves
    6. 2.6 Functions on M
    7. 2.7 Vectors and vector fields
    8. 2.8 Basis vectors and basis vector fields
    9. 2.9 Fiber bundles
    10. 2.10 Examples of fiber bundles
    11. 2.11 A deeper look at fiber bundles
    12. 2.12 Vector fields and integral curves
    13. 2.13 Exponentiation of the operator d/dλ
    14. 2.14 Lie brackets and noncoordinate bases
    15. 2.15 When is a basis a coordinate basis?
    16. 2.16 One-forms
    17. 2.17 Examples of one-forms
    18. 2.18 The Dirac delta function
    19. 2.19 The gradient and the pictorial representation of a one-form
    20. 2.20 Basis one-forms and components of one-forms
    21. 2.21 Index notation
    22. 2.22 Tensors and tensor fields
    23. 2.23 Examples of tensors
    24. 2.24 Components of tensors and the outer product
    25. 2.25 Contraction
    26. 2.26 Basis transformations
    27. 2.27 Tensor operations on components
    28. 2.28 Functions and scalars
    29. 2.29 The metric tensor on a vector space
    30. 2.30 The metric tensor field on a manifold
    31. 2.31 Special relativity
    32. 2.32 Bibliography
  9. 3. Lie derivatives and Lie groups
    1. 3.1 Introduction: how a vector field maps a manifold into itself
    2. 3.2 Lie dragging a function
    3. 3.3 Lie dragging a vector field
    4. 3.4 Lie derivatives
    5. 3.5 Lie derivative of a one-form
    6. 3.6 Submanifolds
    7. 3.7 Frobenius’ theorem (vector field version)
    8. 3.8 Proof of Frobenius’ theorem
    9. 3.9 An example: the generators of S2
    10. 3.10 Invariance
    11. 3.11 Killing vector fields
    12. 3.12 Killing vectors and conserved quantities in particle dynamics
    13. 3.13 Axial symmetry
    14. 3.14 Abstract Lie groups
    15. 3.15 Examples of Lie groups
    16. 3.16 Lie algebras and their groups
    17. 3.17 Realizations and representations
    18. 3.18 Spherical symmetry, spherical harmonics and representations of the rotation group
    19. 3.19 Bibliography
  10. 4. Differential forms
    1. A The algebra and integral calculus of forms
    2. 4.1 Definition of volume – the geometrical role of differential forms
    3. 4.2 Notation and definitions for antisymmetric tensors
    4. 4.3 Differential forms
    5. 4.4 Manipulating differential forms
    6. 4.5 Restriction of forms
    7. 4.6 Fields of forms
    8. 4.7 Handedness and orientability
    9. 4.8 Volumes and integration on oriented manifolds
    10. 4.9 N-vectors, duals, and the symbol εij. . .k
    11. 4.10 Tensor densities
    12. 4.11 Generalized Kronecker deltas
    13. 4.12 Determinants and εij. . .k
    14. 4.13 Metric volume elements
    15. B The differential calculus of forms and its applications
    16. 4.14 The exterior derivative
    17. 4.15 Notation for derivatives
    18. 4.16 Familiar examples of exterior differentiation
    19. 4.17 Integrability conditions for partial differential equations
    20. 4.18 Exact forms
    21. 4.19 Proof of the local exactness of closed forms
    22. 4.20 Lie derivatives of forms
    23. 4.21 Lie derivatives and exterior derivatives commute
    24. 4.22 Stokes’ theorem
    25. 4.23 Gauss’ theorem and the definition of divergence
    26. 4.24 A glance at cohomology theory
    27. 4.25 Differential forms and differential equations
    28. 4.26 Frobenius’ theorem (differential forms version)
    29. 4.27 Proof of the equivalence of the two versions of Frobenius’ theorem
    30. 4.28 Conservation laws
    31. 4.29 Vector spherical harmonics
    32. 4.30 Bibliography
  11. 5. Applications in physics
    1. A Thermodynamics
    2. 5.1 Simple systems
    3. 5.2 Maxwell and other mathematical identities
    4. 5.3 Composite thermodynamic systems: Caratheodory’s theorem
    5. B Hamiltonian mechanics
    6. 5.4 Hamiltonian vector fields
    7. 5.5 Canonical transformations
    8. 5.6 Map between vectors and one-forms provided by ω
    9. 5.7 Poisson bracket
    10. 5.8 Many-particle systems: symplectic forms
    11. 5.9 Linear dynamical systems: the symplectic inner product and conserved quantities
    12. 5.10 Fiber bundle structure of the Hamiltonian equations
    13. C Electromagnetism
    14. 5.11 Rewriting Maxwell’s equations using differential forms
    15. 5.12 Charge and topology
    16. 5.13 The vector potential
    17. 5.14 Plane waves: a simple example
    18. D Dynamics of a perfect fluid
    19. 5.15 Role of Lie derivatives
    20. 5.16 The comoving time-derivative
    21. 5.17 Equation of motion
    22. 5.18 Conservation of vorticity
    23. E Cosmology
    24. 5.19 The cosmological principle
    25. 5.20 Lie algebra of maximal symmetry
    26. 5.21 The metric of a spherically symmetric three-space
    27. 5.22 Construction of the six Killing vectors
    28. 5.23 Open, closed, and flat universes
    29. 5.24 Bibliography
  12. 6. Connections for Riemannian manifolds and gauge theories
    1. 6.1 Introduction
    2. 6.2 Parallelism on curved surfaces
    3. 6.3 The covariant derivative
    4. 6.4 Components: covariant derivatives of the basis
    5. 6.5 Torsion
    6. 6.6 Geodesics
    7. 6.7 Normal coordinates
    8. 6.8 Riemann tensor
    9. 6.9 Geometric interpretation of the Riemann tensor
    10. 6.10 Flat spaces
    11. 6.11 Compatibility of the connection with volume-measure or the metric
    12. 6.12 Metric connections
    13. 6.13 The affine connection and the equivalence principle
    14. 6.14 Connections and gauge theories: the example of electromagnetism
    15. 6.15 Bibliography
  13. Appendix: solutions and hints for selected exercises
  14. Notation
  15. Index