Introduction to Mathematical Physics

This book is intended to be a survey of mathematical methods which should be available to graduate students in physics and related areas in science and engineering at an early stage in their careers. It may also be useful for practicing professionals who need an elementary introduction to the mathematical areas covered in the text.
In addition to the usual topics of analysis such as infinite series, functions of a complex variable, differential and partial differential equations, there is an extensive early introduction to the methods of differential geometry with examples chosen from classical mechanics, thermodynamics and electromagnetic theory. Some simple examples of nonlinear partial differential equations are discussed briefly. There is also a more extensive treatment of group theory than is available in current general mathematical physics textbooks.
1. Infinite Sequences and Series.
2. Finite–Dimensional Vector Spaces.
3. Geometry in Physics.
4. Functions of a Complex Variable.
5. Differential Equations.
6. Hilbert Spaces.
7. Linear Operators on Hilbert Space.
8. Partial Differential Equations.
9. Discrete Groups.
10. Lie Groups and Lie Algebras.

Spis treści:
1 Infinite Sequences and Series.
1.1 Real and Complex Numbers.
1.2 Convergence of Infinite Series and Products.
1.3 Sequences and Series of Functions.
1.4 Asymptotic Series.
2 Finite–Dimensional Vector Spaces.
2.1 Linear Vector Spaces.
2.2 Linear Operators.
2.3 Eigenvectors and Eigenvalues.
2.4 Functions of Operators.
2.5 Linear Dynamical Systems.
3 Geometry in Physics.
3.1 Manifolds andCoordinates.
3.2 Vectors, Differential Forms, and Tensors.
3.3 Calculuson Manifolds.
3.4 Metric Tensor and Distance.
3.5 Dynamical Systems and Vector Fields.
3.6 Fluid Mechanics.
4 Functions of a Complex Variable.
4.1 Elementary Properties of Analytic Functions.
4.2 I ntegration in theComplex Plane.
4.3 Analytic Functions.
4.4 Calculus of Residues: Applications.
4.5 Periodic Functions; Fourier Series.
5 Differential Equations: Analytical Methods.
5.1 Systems of Differential Equations.
5.2 First–Order Differential Equations.
5.3 Linear Differential Equations.
5.4 Linear Second–Order Equations.
5.5 Legendre’s Equation.
5.6 Bessel’s Equation.
6 Hilbert Spaces.
6.1 Infinite–Dimensional Vector Spaces.
6.2 Function Spaces; Measure Theory.
6.3 Fourier Series.
6.4 Fourier Integral; Integral Transforms.
6.5 Orthogonal Polynomials.
6.6 Haar Functions; Wavelets.
7 Linear Operators on Hilbert Space.
7.1 Some Hilbert Space Subtleties.
7.2 General Properties of Linear Operators on Hilbert Space.
7.3 Spectrum of Linear Operators on Hilbert Space.
7.4 Linear Differential Operators.
7.5 Linear Integral Operators; Green Functions.
8 Partial Differential Equations.
8.1 LinearFirst–OrderEquations.
8.2 The Laplacian and Linear Second–Order Equations.
8.3 Time–Dependent Partial Differential Equations.
8.4 Nonlinear Partial Differential Equations.
9 Finite Groups.
9.1 General Properties of Groups.
9.2 Some Finite Groups.
9.3 The Symmetric Group SN.
9.4 Group Representations.
9.5 Representations of the Symmetric Group SN.
9.6 Discrete Infinite Groups.
10 Lie Groups and Lie Algebras.
10.1 Lie Groups.
10.2 Lie Algebras.
10.3 Representationsof Lie Algebras.
Index.

Nota biograficzna:
Michael T. Vaughn is Professor of Physics at Northeastern University in Boston and well known in particle theory for his contributions to quantum field theory especially in the derivation of two loop renormalization group equations for the Yukowa and scalar quartic couplings in Yang–Mills gauge theories and in softly bro ken supersymmetric theories. Professor Vaughn has taught graduate courses in mathematical physics at the University of Pennsylvania, Indiana University and Texas A&M University as well as at Northeastern.

Okładka tylna:
This book is intended to be a survey of mathematical methods which should be available to graduate students in physics and related areas in science and engineering at an early stage in their careers. It may also be useful for practicing professionals who need an elementary introduction to the mathematical areas covered in the text.
In addition to the usual topics of analysis such as infinite series, functions of a complex variable, differential and partial differential equations, there is an extensive early introduction to the methods of differential geometry with examples chosen from classical mechanics, thermodynamics and electromagnetic theory. Some simple examples of nonlinear partial differential equations are discussed briefly. There is also a more extensive treatment of group theory than is available in current general mathematical physics textbooks.
1. Infinite Sequences and Series.
2. Finite–Dimensional Vector Spaces.
3. Geometry in Physics.
4. Functions of a Complex Variable.
5. Differential Equations.
6. Hilbert Spaces.
7. Linear Operators on Hilbert Space.
8. Partial Differential Equations.
9. Discrete Groups.
10. Lie Groups and Lie Algebras.