Textbook in PDF format

This book has been written to help you in your development as an applied scientist, whether engineer, physicist, chemist, or mathematician. It contains material which you will find of great use, not only in the technical courses you have yet to take, but also in your profession after graduation, as long as you deal with the analytical aspects of your field. We have tried to write a book which you will find not only useful but also easy to study, at least as easy as a book on advanced mathematics can be. There is a good deal of theory in it, for it is the theoretical portion of a subject which is the basis for the nonroutine applications of tomorrow. But nowhere will you find theory for its own sake, interesting and legitimate as this may be to a pure mathematician. Our theoretical discussions are designed to illuminate principles, to indicate generalizations, to establish limits within which a given technique may or may not be safely used, or to point out pitfalls into which one might otherwise stumble. On the other hand, there are many applications illustrating, with the material at hand, the usual steps in the solution of a physical problem: formulation, manipulation, and interpretation. These examples are, without exception, carefully set up and completely worked, with all but the simplest steps included. Study them carefully, with paper and pencil at hand, for they are an integral part of the text. If you do this, you should find the exercises, though challenging, still within your ability to work.
Preface
To the Student
Ordinary Differential Equations of the First Order
Functions and Equations
Classification of Differential Equations
Solutions of Differential Equations
Solution Curves and Integral Curves
Differential Equations with Prescribed Solutions
Existence and Uniqueness of Solutions
Exact First-Order Equations
Integrating Factors for First-Order Equations
Separable First-Order Equations
Homogeneous First-Order Equations
Linear First-Order Equations
Special First-Order Equations
Second-Order Equations of Reducible Order
Orthogonal Trajectories
Applications of First-Order Differential Equations
Linear Differential Equations
A Fundamental Existence and Uniqueness Theorem
Families of Solutions
Solutions of Nonhomogeneous Equations
Variation of Parameters and Reduction of Order
Homogeneous Second-Order Equations with Constant Coefficients
Homogeneous Equations of Higher Order
Nonhomogeneous Equations with Constant Coefficients
The Euler-Cauchy Differential Equation
Applications of Linear Differential Equations with Constant Coefficients
Green’s Functions
Introduction to Linear Algebra
The Algebra of Vectors
The Algebra of Matrices
Special Matrices
Determinants
Systems of Linear Algebraic Equations
Special Linear Systems, Inverses, Adjoints, and Cramer’s Rule
Characteristic-Value Problems
Simultaneous Linear Differential Equations
Solutions, Consistency, and Equivalence of Linear Differential Systems
The Reduction of a Differential System to an Equivalent System
Fundamental Concepts and Theorems Concerning First-Order Systems
Complementary Functions and Particular Integrals of Linear Differential Systems
Linear Differential Systems with Constant Coefficients
Finite Differences
The Differences of a Function
Interpolation Formulas
Numerical Differentiation and Integration
The Numerical Solution of Differential Equations
Difference Equations
Difference Equations and the Numerical Solution of Differential Equations
Mechanical Systems and Electric Circuits
Systems with One Degree of Freedom
The Translational Mechanical System
The Series Electric Circuit
Systems with Several Degrees of Freedom
Fourier Series and Integrals
Periodic Functions
The Euler Coefficients
Alternative Formulas for the Fourier Coefficients
Half-Range Expansions
Alternative Forms of Fourier Series
Applications of Fourier Series
The Fourier Integral as the Limit of a Fourier Series
Applications of Fourier Integrals
From the Fourier Integral to the Laplace Transform
The Laplace Transformation
Theoretical Preliminaries
The General Method
The Transforms of Special Functions
Further General Theorems
The Heaviside Expansion Theorems
The Transforms of Periodic Functions
Convolution and the Duhamel Formulas
Partial Differential Equations
The Derivation of Equations
The d’Alembert Solution of the Wave Equation
Characteristics and the Classification of Partial Differential Equations
Separation of Variables
Orthogonal Functions and the General Expansion Problem
Further Applications
Laplace Transform Methods
The Numerical Solution of Partial Differential Equations
Bessel Functions and Legendre Polynomials
Theoretical Preliminaries
The Series Solution of Bessel’s Equation
Modified Bessel Functions
Equations Solvable in Terms of Bessel Functions
Identities for the Bessel Functions
The Orthogonality of the Bessel Functions
Applications of Bessel Functions
Legendre Polynomials
Vector Spaces and Linear Transformations
Vector Spaces
Subspaces, Linear Dependence, and Linear Independence
Bases and Dimension
Linear Transformations
Sums, Products, and Inverses of Linear Transformations
Linear Operator Equations
Applications and Further Properties of Matrices
Transition Probabilities and a Mass-Spring System
Rank and the Equivalence of Matrices
The Existence of Green’s Functions and Their Use in Solving Nonhomogeneous Differential Systems
Quadratic Forms
Characteristic Values and Characteristic Vectors of a Matrix
The Transformation of Matrices
Functions of a Square Matrix
Vector Analysis
The Algebra of Vectors
Vector Functions of One Variable
The Operator V
Line, Surface, and Volume Integrals
Integral Theorems
Further Applications
The Calculus of Variations
Systems of Euler-Lagrange Equations
The Extrema of Integrals under Constraints
Sturm-Liouville Problems
Variations
Hamilton’s Principle and Lagrange Equations of Motion
Analytic Functions of a Complex Variable
Algebraic Preliminaries
The Geometric Representation of Complex Numbers
Absolute Values
Functions of a Complex Variable
Analytic Functions
The Elementary Functions of z
Integration in the Complex Plane
Analytic Functions and Two-Dimensional Field Theory
Infinite Series in the Complex Plane
Series of Complex Terms
Taylor’s Expansion
Laurent’s Expansion
The Theory of Residues
The Residue Theorem
The Evaluation of Real Definite Integrals
The Complex Inversion Integral
Stability Criteria
Conformal Mapping
The Geometrical Representation of Functions of z
Conformal Mapping
The Bilinear Transformation
The Schwarz-Christoffel Transformation
Answers to Odd-Numbered Exercises
Index