Introduction To Mathematical Physics

A comprehensive survey of all the mathematical methods that should be available to graduate students in physics. In addition to the usual topics of analysis, such as infinite series, functions of a complex variable and some differential equations as well as linear vector spaces, this book includes a more extensive discussion of group theory than can be found in other current textbooks. The main feature of this textbook is its extensive treatment of geometrical methods as applied to physics. With its introduction of differentiable manifolds and a discussion of vectors and forms on such manifolds as part of a first-year graduate course in mathematical methods, the text allows students to grasp at an early stage the contemporary literature on dynamical systems, solitons and related topological solutions to field equations, gauge theories, gravitational theory, and even string theory.
Free solutions manual available for lecturers at www.wiley-vch.de/supplements/.

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 broken 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.

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 Integration 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.