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A Course in Modern Mathematical Physics

Book Description

This book provides an introduction to the major mathematical structures used in physics today. It covers the concepts and techniques needed for topics such as group theory, Lie algebras, topology, Hilbert space and differential geometry. Important theories of physics such as classical and quantum mechanics, thermodynamics, and special and general relativity are also developed in detail, and presented in the appropriate mathematical language. The book is suitable for advanced undergraduate and beginning graduate students in mathematical and theoretical physics, as well as applied mathematics. It includes numerous exercises and worked examples, to test the reader's understanding of the various concepts, as well as extending the themes covered in the main text. The only prerequisites are elementary calculus and linear algebra. No prior knowledge of group theory, abstract vector spaces or topology is required.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface
  6. Acknowledgements
  7. Dedication
  8. 1 Sets and structures
    1. 1.1 Sets and logic
    2. 1.2 Subsets, unions and intersections of sets
    3. 1.3 Cartesian products and relations
    4. 1.4 Mappings
    5. 1.5 Infinite sets
    6. 1.6 Structures
    7. 1.7 Category theory
  9. 2 Groups
    1. 2.1 Elements of group theory
    2. 2.2 Transformation and permutation groups
    3. 2.3 Matrix groups
    4. 2.4 Homomorphisms and isomorphisms
    5. 2.5 Normal subgroups and factor groups
    6. 2.6 Group actions
    7. 2.7 Symmetry groups
  10. 3 Vector spaces
    1. 3.1 Rings and fields
    2. 3.2 Vector spaces
    3. 3.3 Vector space homomorphisms
    4. 3.4 Vector subspaces and quotient spaces
    5. 3.5 Bases of a vector space
    6. 3.6 Summation convention and transformation of bases
    7. 3.7 Dual spaces
  11. 4 Linear operators and matrices
    1. 4.1 Eigenspaces and characteristic equations
    2. 4.2 Jordan canonical form
    3. 4.3 Linear ordinary differential equations
    4. 4.4 Introduction to group representation theory
  12. 5 Inner product spaces
    1. 5.1 Real inner product spaces
    2. 5.2 Complex inner product spaces
    3. 5.3 Representations of finite groups
  13. 6 Algebras
    1. 6.1 Algebras and ideals
    2. 6.2 Complex numbers and complex structures
    3. 6.3 Quaternions and Clifford algebras
    4. 6.4 Grassmann algebras
    5. 6.5 Lie algebras and Lie groups
  14. 7 Tensors
    1. 7.1 Free vector spaces and tensor spaces
    2. 7.2 Multilinear maps and tensors
    3. 7.3 Basis representation of tensors
    4. 7.4 Operations on tensors
  15. 8 Exterior algebra
    1. 8.1 r-Vectors and r-forms
    2. 8.2 Basis representation of r-vectors
    3. 8.3 Exterior product
    4. 8.4 Interior product
    5. 8.5 Oriented vector spaces
    6. 8.6 The Hodge dual
  16. 9 Special relativity
    1. 9.1 Minkowski space-time
    2. 9.2 Relativistic kinematics
    3. 9.3 Particle dynamics
    4. 9.4 Electrodynamics
    5. 9.5 Conservation laws and energy–stress tensors
  17. 10 Topology
    1. 10.1 Euclidean topology
    2. 10.2 General topological spaces
    3. 10.3 Metric spaces
    4. 10.4 Induced topologies
    5. 10.5 Hausdorff spaces
    6. 10.6 Compact spaces
    7. 10.7 Connected spaces
    8. 10.8 Topological groups
    9. 10.9 Topological vector spaces
  18. 11 Measure theory and integration
    1. 11.1 Measurable spaces and functions
    2. 11.2 Measure spaces
    3. 11.3 Lebesgue integration
  19. 12 Distributions
    1. 12.1 Test functions and distributions
    2. 12.2 Operations on distributions
    3. 12.3 Fourier transforms
    4. 12.4 Green’s functions
  20. 13 Hilbert spaces
    1. 13.1 Definitions and examples
    2. 13.2 Expansion theorems
    3. 13.3 Linear functionals
    4. 13.4 Bounded linear operators
    5. 13.5 Spectral theory
    6. 13.6 Unbounded operators
  21. 14 Quantum mechanics
    1. 14.1 Basic concepts
    2. 14.2 Quantum dynamics
    3. 14.3 Symmetry transformations
    4. 14.4 Quantum statistical mechanics
  22. 15 Differential geometry
    1. 15.1 Differentiable manifolds
    2. 15.2 Differentiable maps and curves
    3. 15.3 Tangent, cotangent and tensor spaces
    4. 15.4 Tangent map and submanifolds
    5. 15.5 Commutators, flows and Lie derivatives
    6. 15.6 Distributions and Frobenius theorem
  23. 16 Differentiable forms
    1. 16.1 Differential forms and exterior derivative
    2. 16.2 Properties of exterior derivative
    3. 16.3 Frobenius theorem: dual form
    4. 16.4 Thermodynamics
    5. 16.5 Classical mechanics
  24. 17 Integration on manifolds
    1. 17.1 Partitions of unity
    2. 17.2 Integration of n-forms
    3. 17.3 Stokes’ theorem
    4. 17.4 Homology and cohomology
    5. 17.5 The Poincaré lemma
  25. 18 Connections and curvature
    1. 18.1 Linear connections and geodesics
    2. 18.2 Covariant derivative of tensor fields
    3. 18.3 Curvature and torsion
    4. 18.4 Pseudo-Riemannian manifolds
    5. 18.5 Equation of geodesic deviation
    6. 18.6 The Riemann tensor and its symmetries
    7. 18.7 Cartan formalism
    8. 18.8 General relativity
    9. 18.9 Cosmology
    10. 18.10 Variation principles in space-time
  26. 19 Lie groups and Lie algebras
    1. 19.1 Lie groups
    2. 19.2 The exponential map
    3. 19.3 Lie subgroups
    4. 19.4 Lie groups of transformations
    5. 19.5 Groups of isometries
  27. Bibliography
  28. Index