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Mathematics for Physics

Book Description

An engagingly-written account of mathematical tools and ideas, this book provides a graduate-level introduction to the mathematics used in research in physics. The first half of the book focuses on the traditional mathematical methods of physics - differential and integral equations, Fourier series and the calculus of variations. The second half contains an introduction to more advanced subjects, including differential geometry, topology and complex variables. The authors' exposition avoids excess rigor whilst explaining subtle but important points often glossed over in more elementary texts. The topics are illustrated at every stage by carefully chosen examples, exercises and problems drawn from realistic physics settings. These make it useful both as a textbook in advanced courses and for self-study. Password-protected solutions to the exercises are available to instructors at www.cambridge.org/9780521854030.

Table of Contents

  1. Cover
  2. Half Title
  3. Title Page
  4. Copyright
  5. Dedication
  6. Contents
  7. Preface
  8. Acknowledgments
  9. 1. Calculus of variations
    1. 1.1 What is it good for?
    2. 1.2 Functionals
    3. 1.3 Lagrangian mechanics
    4. 1.4 Variable endpoints
    5. 1.5 Lagrange multipliers
    6. 1.6 Maximum or minimum?
    7. 1.7 Further exercises and problems
  10. 2. Function spaces
    1. 2.1 Motivation
    2. 2.2 Norms and inner products
    3. 2.3 Linear operators and distributions
    4. 2.4 Further exercises and problems
  11. 3. Linear ordinary differential equations
    1. 3.1 Existence and uniqueness of solutions
    2. 3.2 Normal form
    3. 3.3 Inhomogeneous equations
    4. 3.4 Singular points
    5. 3.5 Further exercises and problems
  12. 4. Linear differential operators
    1. 4.1 Formal vs. concrete operators
    2. 4.2 The adjoint operator
    3. 4.3 Completeness of eigenfunctions
    4. 4.4 Further exercises and problems
  13. 5. Green functions
    1. 5.1 Inhomogeneous linear equations
    2. 5.2 Constructing Green functions
    3. 5.3 Applications of Lagrange’s identity
    4. 5.4 Eigenfunction expansions
    5. 5.5 Analytic properties of Green functions
    6. 5.6 Locality and the Gelfand–Dikii equation
    7. 5.7 Further exercises and problems
  14. 6. Partial differential equations
    1. 6.1 Classification of PDEs
    2. 6.2 Cauchy data
    3. 6.3 Wave equation
    4. 6.4 Heat equation
    5. 6.5 Potential theory
    6. 6.6 Further exercises and problems
  15. 7. The mathematics of real waves
    1. 7.1 Dispersive waves
    2. 7.2 Making waves
    3. 7.3 Nonlinear waves
    4. 7.4 Solitons
    5. 7.5 Further exercises and problems
  16. 8. Special functions
    1. 8.1 Curvilinear coordinates
    2. 8.2 Spherical harmonics
    3. 8.3 Bessel functions
    4. 8.4 Singular endpoints
    5. 8.5 Further exercises and problems
  17. 9. Integral equations
    1. 9.1 Illustrations
    2. 9.2 Classification of integral equations
    3. 9.3 Integral transforms
    4. 9.4 Separable kernels
    5. 9.5 Singular integral equations
    6. 9.6 Wiener–Hopf equations I
    7. 9.7 Some functional analysis
    8. 9.8 Series solutions
    9. 9.9 Further exercises and problems
  18. 10. Vectors and tensors
    1. 10.1 Covariant and contravariant vectors
    2. 10.2 Tensors
    3. 10.3 Cartesian tensors
    4. 10.4 Further exercises and problems
  19. 11. Differential calculus on manifolds
    1. 11.1 Vector and covector fields
    2. 11.2 Differentiating tensors
    3. 11.3 Exterior calculus
    4. 11.4 Physical applications
    5. 11.5 Covariant derivatives
    6. 11.6 Further exercises and problems
  20. 12. Integration on manifolds
    1. 12.1 Basic notions
    2. 12.2 Integrating p-forms
    3. 12.3 Stokes’ theorem
    4. 12.4 Applications
    5. 12.5 Further exercises and problems
  21. 13. An introduction to differential topology
    1. 13.1 Homeomorphism and diffeomorphism
    2. 13.2 Cohomology
    3. 13.3 Homology
    4. 13.4 De Rham’s theorem
    5. 13.5 Poincaré duality
    6. 13.6 Characteristic classes
    7. 13.7 Hodge theory and the Morse index
    8. 13.8 Further exercises and problems
  22. 14. Groups and group representations
    1. 14.1 Basic ideas
    2. 14.2 Representations
    3. 14.3 Physics applications
    4. 14.4 Further exercises and problems
  23. 15. Lie groups
    1. 15.1 Matrix groups
    2. 15.2 Geometry of SU(2)
    3. 15.3 Lie algebras
    4. 15.4 Further exercises and problems
  24. 16. The geometry of fibre bundles
    1. 16.1 Fibre bundles
    2. 16.2 Physics examples
    3. 16.3 Working in the total space
  25. 17. Complex analysis
    1. 17.1 Cauchy–Riemann equations
    2. 17.2 Complex integration: Cauchy and Stokes
    3. 17.3 Applications
    4. 17.4 Applications of Cauchy’s theorem
    5. 17.5 Meromorphic functions and the winding number
    6. 17.6 Analytic functions and topology
    7. 17.7 Further exercises and problems
  26. 18. Applications of complex variables
    1. 18.1 Contour integration technology
    2. 18.2 The Schwarz reflection principle
    3. 18.3 Partial-fraction and product expansions
    4. 18.4 Wiener–Hopf equations II
    5. 18.5 Further exercises and problems
  27. 19. Special functions and complex variables
    1. 19.1 The Gamma function
    2. 19.2 Linear differential equations
    3. 19.3 Solving ODEs via contour integrals
    4. 19.4 Asymptotic expansions
    5. 19.5 Elliptic functions
    6. 19.6 Further exercises and problems
  28. A. Linear algebra review
    1. A.1 Vector space
    2. A.2 Linear maps
    3. A.3 Inner-product spaces
    4. A.4 Sums and differences of vector spaces
    5. A.5 Inhomogeneous linear equations
    6. A.6 Determinants
    7. A.7 Diagonalization and canonical forms
  29. B. Fourier series and integrals
    1. B.1 Fourier series
    2. B.2 Fourier integral transforms
    3. B.3 Convolution
    4. B.4 The Poisson summation formula
  30. References
  31. Index