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Mathematical Methods for Scientists and Engineers

Donald A. McQuarrie University of California, Davis

Detailed Table of Contents

Chapter 1:  Functions of a Single Variable
     1-1.           Functions
     1-2.           Limits
     1-3.           Continuity
     1-4.           Differentiation
     1-5.           Differentials
     1-6.           Mean Value Theorems
     1-7.           Integration
     1-8.           Improper Integrals
     1-9.           Uniform Convergence of Integrals

Chapter 2:  Infinite Series
     2-1.           Infinite Sequences
     2-2.           Convergence and Divergence of Infinite Series
     2-3.           Tests for Convergence
     2-4.           Alternating Series
     2-5.           Uniform Convergence
     2-6.           Power Series
     2-7.           Taylor Series
     2-8.           Applications of Taylor Series
     2-9.           Asymptotic Expansions

Chapter 3:  Functions Defined As Integrals
     3-1.           The Gamma Function
     3-2.           The Beta Function
     3-3.           The Error Function
     3-4.           The Exponential Integral
     3-5.           Elliptic Integrals
     3-6.           The Dirac Delta Function
     3-7.           Bernoulli Numbers and Bernoulli Polynomials

Chapter 4:  Complex Numbers and Complex Functions
     4-1.           Complex Numbers and the Complex Plane
     4-2.           Functions of a Complex Variable
     4-3.           Euler’s Formula and the Polar Form of Complex Numbers
     4-4.           Trigonometric and Hyperbolic Functions
     4-5.            The Logarithms of Complex Numbers
     4-6.           Powers of Complex Numbers

Chapter 5:  Vectors
     5-1.           Vectors in Two Dimensions
     5-2.           Vector Functions in Two Dimensions
     5-3.           Vectors in Three Dimensions
     5-4.           Vector Functions in Three Dimensions
     5-5.           Lines and Planes in Space

Chapter 6:  Functions of Several Variables
     6-1.           Functions
     6-2.           Limits and Continuity
     6-3.           Partial Derivatives
     6-4.           Chain Rules for Partial Differentiation
     6-5.           Differentials and the Total Differential
     6-6.           The Directional Derivative and the Gradient
     6-7.           Taylor’s Formula in Several Variables
     6-8.           Maxima and Minima
     6-9.           The Method of Lagrange Multipliers
     6-10.          Multiple Integrals

Chapter 7:  Vector Calculus
     7-1.           Vector Fields
     7-2.           Line Integrals
     7-3.           Surface Integrals
     7-4.           The Divergence Theorem
     7-5.           Stokes’s Theorem

Chapter 8:  Curvilinear Coordinates
     8-1.           Plane Polar Coordinates
     8-2.           Vectors in Plane Polar Coordinates
     8-3.           Cylindrical Coordinates
     8-4.           Spherical Coordinates
     8-5.           Curvilinear Coordinates
     8-6.           Some Other Coordinate Systems

Chapter 9:  Linear Algebra and Vector Spaces
     9-1.           Determinants
     9-2.           Gaussian Elimination
     9-3.           Matrices
     9-4.           Rank of a Matrix
     9-5.           Vector Spaces
     9-6.           Inner Product Spaces
     9-7.           Complex Inner Product Spaces

Chapter 10:  Matrices and Eigenvalue Problems
     10-1.           Orthogonal and Unitary Transformations
     10-2.           Eigenvalues and Eigenvectors
     10-3.           Some Applied Eigenvalue Problems
     10-4.           Change of Basis
     10-5.           Diagonalization of Matrices
     10-6.           Quadratic Forms

Chapter 11:  Ordinary Differential Equations
     11-1.           Differential Equations of First Order and First Degree
     11-2.           Linear First-Order Differential Equations
     11-3.           Homogeneous Linear Differential Equations with Constant Coefficients
     11-4.           Nonhomogeneous Linear Differential Equations with Constant Coefficients
     11-5.           Some Other Types of Higher-Order Differential Equations
     11-6.           Systems of First-Order Differential Equations
     11-7.           Two Invaluable Resources for Solutions to Differential Equations

Chapter 12:  Series Solutions of Differential Equations
     12-1.          The Power Series Method
     12-2.           Ordinary Points and Singular Points of Differential Equations
     12-3.           Series Solutions Near an Ordinary Point: Legendre’s Equation
     12-4.           Solutions Near Regular Singular Points
     12-5.           Bessel’s Equation
     12-6.           Bessel Functions

Chapter 13:  Qualitative Methods for Nonlinear Differential Equations
     13-1.          The Phase Plane
     13-2.           Critical Points in the Phase Plane
     13-3.           Stability of Critical Points
     13-4.           Nonlinear Oscillators
     13-5.           Population Dynamics

Chapter 14:  Orthogonal Polynomials and Sturm–Liouville Problems
     14-1.           Legendre Polynomials
     14-2.           Orthogonal Polynomials
     14-3.           Sturm–Liouville Theory
     14-4.           Eigenfunction Expansions
     14-5.           Green’s Functions

Chapter 15:  Fourier Series
     15-1.           Fourier Series as Eigenfunction Expansions
     15-2.           Sine and Cosine Series
     15-3.           Convergence of Fourier Series
     15-4.           Fourier Series and Ordinary Differential Equations

Chapter 16:  Partial Differential Equations
     16-1.           Some Examples of Partial Differential Equations
     16-2.           Laplace’s Equation
     16-3.          The One-Dimensional Wave Equation
     16-4.          The Two-Dimensional Wave Equation
     16-5.          The Heat Equation
     16-6.          The Schrödinger Equation
          a.  Particle in a Box
          b.  A Rigid Rotor
          c.  The Electron in a Hydrogen Atom
     16-7.         The Classification of Partial Differential Equations

Chapter 17:  Integral Transforms
     17-1.          The Laplace Transform
     17-2.          The Inversion of Laplace Transforms
     17-3.           Laplace Transforms and Ordinary Differential Equations
     17-4.           Laplace Transforms and Partial Differential Equations
     17-5.           Fourier Transforms
     17-6.           Fourier Transforms and Partial Differential Equations
     17-7.          The Inversion Formula for Laplace Transforms

Chapter 18:  Functions of a Complex Variable: Theory
     18-1.           Functions, Limits, and Continuity
     18-2.           Differentiation. The Cauchy–Riemann Equations
     18-3.           Complex Integration. Cauchy’s Theorem
     18-4.           Cauchy’s Integral Formula
     18-5.           Taylor Series and Laurent Series
     18-6.           Residues and the Residue Theorem

Chapter 19:  Functions of a Complex Variable: Applications
     19-1.          The Inversion Formula for Laplace Transforms
     19-2.           Evaluation of Real, Definite Integrals
     19-3.           Summation of Series
     19-4.           Location of Zeros
     19-5.           Conformal Mapping
     19-6.           Conformal Mapping and Boundary Value Problems
     19-7.           Conformal Mapping and Fluid Flow

Chapter 20:  Calculus of Variations
     20-1.          The Euler’s Equation
     20-2.           Two Laws of Physics in Variational Form
     20-3.           Variational Problems with Constraints
     20-4.           Variational Formulation of Eigenvalue Problems
     20-5.           Multidimensional Variational Problems

Chapter 21:  Probability Theory and Stochastic Processes
     21-1.           Discrete Random Variables
     21-2.           Continuous Random Variables
     21-3.           Characteristic Functions
     21-4.           Stochastic Processes—General
     21-5.           Stochastic Processes—Examples
          a.  Poisson Process
          b.  The Shot Effect

Chapter 22:  Mathematical Statistics
     22-1.           Estimation of Parameters
     22-2.           Three Key Distributions Used in Statistical Tests
          a.     The Normal Distribution
          b.     The Chi–Square Distribution
          c.     Student t-Distribution
     22-3.           Confidence Intervals
          a.  Confidence Intervals for the Mean of a Normal Distribution Whose Variance is Known
          b.  Confidence Intervals for the Mean of a Normal Distribution with Unknown Variance
          c.  Confidence Intervals for the Variance of a Normal Distribution
     22-4.           Goodness of Fit
     22-5.           Regression and Correlation