Cover

Half Title

Title Page

Copyright

Dedication

Contents

Preface

Acknowledgments

1 Calculus of variations

1.1 What is it good for?

1.2 Functionals

1.3 Lagrangian mechanics

1.4 Variable endpoints

1.5 Lagrange multipliers

1.6 Maximum or minimum?

1.7 Further exercises and problems

2 Function spaces

2.1 Motivation

2.2 Norms and inner products

2.3 Linear operators and distributions

2.4 Further exercises and problems

3 Linear ordinary differential equations

3.1 Existence and uniqueness of solutions

3.2 Normal form

3.3 Inhomogeneous equations

3.4 Singular points

3.5 Further exercises and problems

4 Linear differential operators

4.1 Formal vs. concrete operators

4.2 The adjoint operator

4.3 Completeness of eigenfunctions

4.4 Further exercises and problems

5 Green functions

5.1 Inhomogeneous linear equations

5.2 Constructing Green functions

5.3 Applications of Lagrange’s identity

5.4 Eigenfunction expansions

5.5 Analytic properties of Green functions

5.6 Locality and the Gelfand–Dikii equation

5.7 Further exercises and problems

6 Partial differential equations

6.1 Classification of PDEs

6.2 Cauchy data

6.3 Wave equation

6.4 Heat equation

6.5 Potential theory

6.6 Further exercises and problems

7 The mathematics of real waves

7.1 Dispersive waves

7.2 Making waves

7.3 Nonlinear waves

7.4 Solitons

7.5 Further exercises and problems

8 Special functions

8.1 Curvilinear coordinates

8.2 Spherical harmonics

8.3 Bessel functions

8.4 Singular endpoints

8.5 Further exercises and problems

9 Integral equations

9.1 Illustrations

9.2 Classification of integral equations

9.3 Integral transforms

9.4 Separable kernels

9.5 Singular integral equations

9.6 Wiener–Hopf equations I

9.7 Some functional analysis

9.8 Series solutions

9.9 Further exercises and problems

10 Vectors and tensors

10.1 Covariant and contravariant vectors

10.2 Tensors

10.3 Cartesian tensors

10.4 Further exercises and problems

11 Differential calculus on manifolds

11.1 Vector and covector fields

11.2 Differentiating tensors

11.3 Exterior calculus

11.4 Physical applications

11.5 Covariant derivatives

11.6 Further exercises and problems

12 Integration on manifolds

12.1 Basic notions

12.2 Integrating p-forms

12.3 Stokes’ theorem

12.4 Applications

12.5 Further exercises and problems

13 An introduction to differential topology

13.1 Homeomorphism and diffeomorphism

13.2 Cohomology

13.3 Homology

13.4 De Rham’s theorem

13.5 Poincaré duality

13.6 Characteristic classes

13.7 Hodge theory and the Morse index

13.8 Further exercises and problems

14 Groups and group representations

14.1 Basic ideas

14.2 Representations

14.3 Physics applications

14.4 Further exercises and problems

15 Lie groups

15.1 Matrix groups

15.2 Geometry of SU(2)

15.3 Lie algebras

15.4 Further exercises and problems

16 The geometry of fibre bundles

16.1 Fibre bundles

16.2 Physics examples

16.3 Working in the total space

17 Complex analysis

17.1 Cauchy–Riemann equations

17.2 Complex integration: Cauchy and Stokes

17.3 Applications

17.4 Applications of Cauchy’s theorem

17.5 Meromorphic functions and the winding number

17.6 Analytic functions and topology

17.7 Further exercises and problems

18 Applications of complex variables

18.1 Contour integration technology

18.2 The Schwarz reflection principle

18.3 Partial-fraction and product expansions

18.4 Wiener–Hopf equations II

18.5 Further exercises and problems

19 Special functions and complex variables

19.1 The Gamma function

19.2 Linear differential equations

19.3 Solving ODEs via contour integrals

19.4 Asymptotic expansions

19.5 Elliptic functions

19.6 Further exercises and problems

A Linear algebra review

A.1 Vector space

A.2 Linear maps

A.3 Inner-product spaces

A.4 Sums and differences of vector spaces

A.5 Inhomogeneous linear equations

A.6 Determinants

A.7 Diagonalization and canonical forms

B Fourier series and integrals

B.1 Fourier serie

B.2 Fourier integral transforms

B.3 Convolution

B.4 The Poisson summation formula

References

Index