Synopses & Reviews
Intended for college-level physics, engineering, or mathematics students, this volume offers an algebraically based approach to various topics in applied math. It is accessible to undergraduates with a good course in calculus which includes infinite series and uniform convergence. Exercises follow each chapter to test the student's grasp of the material; however, the author has also included exercises that extend the results to new situations and lay the groundwork for new concepts to be introduced later. A list of references for further reading will be found at the end of each chapter.
For this second revised edition, Professor Dettman included a new section on generalized functions to help explain the use of the Dirac delta function in connection with Green's functions. In addition, a new approach to series solutions of ordinary differential equations has made the treatment independent of complex variable theory. This means that the first six chapters can be grasped without prior knowledge of complex variables. However, since Chapter 8 depends heavily on analytic functions of a complex variable, a new Chapter 7 on analytic function theory has been written.
Synopsis
Algebraically based approach to vectors, mapping, diffraction, and other topics covers generalized functions, analytic function theory, Hilbert spaces, calculus of variations, boundary value problems, integral equations, more. 1969 edition.
Synopsis
Algebraically based approach to vectors, mapping, diffraction, and other topics covers generalized functions, analytic function theory, Hilbert spaces, calculus of variations, boundary value problems, integral equations, more. 1969 edition.
Table of Contents
Preface
CHAPTER 1. Linear Algebra
1.1 Linear Equations. Summation Convention
1.2 Matrices
1.3 Determinants
1.4 Systems of Linear Algebraic Equations. Rank of a Matrix
1.5 Vector Spaces
1.6 Scalar Product
1.7 Orthonormal Basis. Linear Transformations
1.8 Quadratic Forms. Hermitian Forms
1.9 Systems of Ordinary Differential Equations. Vibration Problems
1.10 Linear Programming
CHAPTER 2. Hilbert Spaces
2.1 Infinite-dimensional Vector Spaces. Function Spaces
2.2 Fourier Series
2.3 Separable Hilbert Spaces
2.4 The Projection Theorem
2.5 Linear Functionals
2.6 Weak Convergence
2.7 Linear Operators
2.8 Completely Continuous Operators
CHAPTER 3. Calculus of Variations
3.1 Maxima and Minima of Functions. Lagrange Multipliers
3.2 Maxima and Minima of Functionals. Euler's Equation
3.3 Hamilton's Principle. Lagrange's Equations
3.4 Theory of Small Vibrations
3.5 The Vibrating String
3.6 Boundary-value Problems of Mathematical Physics
3.7 Eigenvalues and Eigenfunctions
3.8 Eigenfunction Expansions
3.9 Upper and Lower Bounds for Eigenvalues
CHAPTER 4. Boundary-value Problems. Separation of Variables
4.1 Orthogonal Coordinate Systems. Separation of Variables
4.2 Sturm-Liouville Problems
4.3 Series Solutions of Ordinary Differential Equations
4.4 Series Solutions of Boundary-value Problems
CHAPTER 5. Boundary-value Problems. Green's Functions
5.1 Nonhomogeneous Boundary-value Problems
5.2 One-dimensional Green's Functions
5.3 Generalized Functions
5.4 Green's Functions in Higher Dimensions
5.5 Problems in Unbounded Regions
5.6 A Problem in Diffraction Theory
CHAPTER 6. Integral Equations
6.1 Integral-equation Formulation of Boundary-value Problems
6.2 Hilbert-Schmidt Theory
6.3 Fredholm Theory
6.4 Integral Equations of the First Kind
CHAPTER 7. Analytic Function Theory
7.1 Introduction
7.2 Analytic Functions
7.3 Elementary Functions
7.4 Complex Integration
7.5 Integral Representations
7.6 Sequences and Series
7.7 Series Representations of Analytic Functions
7.8 Contour Integration
7.9 Conformal Mapping
7.10 Potential Theory
CHAPTER 8. Integral Transform Methods
8.1 Fourier Transforms
8.2 Applications of Fourier Transforms. Ordinary Differential Equations
8.3 Applications of Fourier Transforms. Partial Differential Equations
8.4 Applications of Fourier Transforms. Integral Equations
8.5 Laplace Transforms. Applications
8.6 Other Transform Techniques
Index