Chapter 1: Infinite Series, Power SeriesSection 1:
The Geometric Series
Section 2:
Definitions And Notation
Section 4:
Convergent And Divergent Series
Section 5:
Testing Series For Convergence; The Preliminary Test
Section 6:
Convergence Tests For Series Of Positive Terms; Absolute Convergence
Section 7:
Alternating Series
Section 9:
Useful Facts About Series
Section 10:
Power Series; Interval Of Convergence
Section 12:
Expanding Functions In Power Series
Section 13:
Techniques For Obtaining Power Series Expansions
Section 14:
Accuracy Of Series Approximations
Section 15:
Some Uses Of Series
Section 16:
Miscellaneous Problems
Chapter 2: Complex NumbersSection 4:
Terminology And Notation
Section 5:
Complex Algebra
Section 6:
Complex Infinite Series
Section 7:
Complex Power Series; Disk of Convergence
Section 8:
Elementary Functions of Complex Numbers
Section 9:
Euler’s Formula
Section 10:
Powers and Roots of Complex Numbers
Section 11:
The Exponential and Trigonometric Functions
Section 12:
Hyperbolic Functions
Section 14:
Complex Roots and Powers
Section 15:
Inverse Trigonometric and Hyperbolic Functions
Section 16:
Some Applications
Section 17:
Miscellaneous Problems
Section 2:
Matrices; Row Reduction
Section 3:
Determinants; Cramer’s Rule
Section 4:
Vectors
Section 5:
Lines and Planes
Section 6:
Complex Infinite Series
Section 7:
Linear Combinations, Linear Functions, Linear Operators
Section 8:
Linear Dependence and Independence
Section 9:
Special Matrices and Formulas
Section 10:
Linear Vector Spaces
Section 11:
Eigenvalues and Eigenvectors; Diagonalizing Matrices
Section 12:
Applications of Diagonalization
Section 13:
A Brief Introduction to Groups
Section 14:
General Vector Spaces
Section 15:
Miscellaneous Problems
Chapter 4: Partial DifferentiationSection 1:
Introduction and Notation
Section 2:
Power Series in Two Variables
Section 3:
Total DifferentialsSection 4:
Approximations using Differentials
Section 5:
Chain Rule or Differentiating a Function of a Function
Section 6:
Implicit Differentiation
Section 7:
More Chain Rule
Section 8:
Application of Partial Differentiation to Maximum and Minimum Problems
Section 9:
Maximum and Minimum Problems with Constraints; Lagrange Multipliers
Section 10:
Endpoint or Boundary Point Problems
Section 11:
Change of Variables
Section 12:
Differentiation of Integrals; Leibniz’ Rule
Section 13:
Miscellaneous Problems
Chapter 5: Multiple Integrals; Applications of IntegrationSection 1:
Introduction
Section 2:
Double and Triple Integrals
Section 3:
Applications of Integration; Single and Multiple Integrals
Section 4:
Change of Variables in Integrals; Jacobian's
Section 5:
Surface Integrals
Section 6:
Miscellaneous Problems
Chapter 6: Vector AnalysisSection 3:
Triple Products
Section 4:
Differentiation of Vectors
Section 6:
Directional Derivative; Gradient
Section 7:
Some other Expressions Involving ∇
Section 8:
Line Integrals
Section 9:
Green’s Theorem in the Plane
Section 10:
The Divergence and The Divergence Theorem
Section 11:
The Curl and Stokes’ Theorem
Section 12:
Miscellaneous Problems
Chapter 7: Fourier Series and TransformsSection 2:
Simple Harmonic Motion and Wave Motion; Periodic Functions
Section 3:
Applications of Fourier Series
Section 4:
Average Value of a Function
Section 5:
Fourier Coefficients
Section 6:
Dirichlet Conditions
Section 7:
Complex Form of Fourier Series
Section 8:
Other Intervals
Section 9:
Even and Odd Functions
Section 10:
An Application to Sound
Section 11:
Parseval’s Theorem
Section 12:
Fourier Transforms
Section 13:
Miscellaneous Problems
Chapter 8: Ordinary Differential EquationsSection 1:
Introduction
Section 2:
Separable Equations
Section 3:
Linear First-Order Equations
Section 4:
Other Methods for First-Order Equations
Section 5:
Second-Order Linear Equations with Constant Coefficients and Zero Right-Hand Side
Section 6:
Second-Order Linear Equations with Constant Coefficients and Right-Hand Side Not Zero
Section 7:
Other Second-Order Equations
Section 8:
The Laplace Transform
Section 9:
Solution of Differential Equations by Laplace Transforms
Section 10:
Convolution
Section 11:
The Dirac Delta Function
Section 12:
A Brief Introduction to Green Functions
Section 13:
Miscellaneous Problems
Chapter 9: Calculus of VariationsSection 1:
Introduction
Section 2:
The Euler Equation
Section 3:
Using the Euler Equation
Section 4:
The Brachistochrone Problem; Cycloids
Section 5:
Several Dependent Variables; Lagrange’s Equations
Section 6:
Isoperimetric Problems
Section 8:
Miscellaneous Problems
Chapter 10: Tensor AnalysisSection 2:
Cartesian Tensors
Section 3:
Tensor Notation and Operations
Section 4:
Inertia Tensor
Section 5:
Kronecker Delta and Levi-Civita Symbol
Section 6:
Pseudovectors and Pseudotensors
Section 7:
More about Applications
Section 8:
Curvilinear Coordinates
Section 9:
Vector Operators in Orthogonal Curvilinear Coordinates
Section 10:
Non-Cartesian Tensors
Section 11:
Miscellaneous Problems
Chapter 11: Special FunctionsSection 2:
The Factorial Function
Section 3:
Definition of the Gamma Function; Recursion Relation
Section 5:
Some Important Formulas Involving Gamma Functions
Section 6:
Beta Functions
Section 7:
Beta Functions in Terms of Gamma Functions
Section 8:
The Simple Pendulum
Section 9:
The Error Function
Section 10:
Asymptotic Series
Section 11:
Stirling’s Formula
Section 12:
Elliptic Integrals and Functions
Section 13:
Miscellaneous Problems
Exercise 1
Exercise 2
Exercise 3
Exercise 4
Chapter 12: Series Solutions of Differential Equations; Legendre, Bessel, Hermite, and Laguerre FunctionsSection 1:
Introduction
Section 2:
Legendre’s Equation
Section 3:
Leibniz’ Rule for Differentiating Products
Section 4:
Rodrigues’ Formula
Section 5:
Generating Function for Legendre Polynomials
Section 6:
Complete Sets of Orthogonal Functions
Section 7:
Orthogonality of The Legendre Polynomials
Section 8:
Normalization of The Legendre Polynomials
Section 9:
Legendre Series
Section 10:
The Associated Legendre Functions
Section 11:
Generalized Power Series or The Method of Frobenius
Section 12:
Bessel’s Equation
Section 13:
The Second Solution of Bessel’S Equation
Section 14:
Graphs and Zeros of Bessel Functions
Section 15:
Recursion Relations
Section 16:
Differential Equations with Bessel Function Solutions
Section 17:
Other Kinds of Bessel Functions
Section 18:
The Lengthening Pendulum
Section 19:
Orthogonality of Bessel Functions
Section 20:
Approximate Formulas for Bessel Functions
Section 21:
Series Solutions; Fuchs’S Theorem
Section 22:
Hermite Functions; Laguerre Functions; Ladder Operators
Section 23:
Miscellaneous Problems
Chapter 13: Partial Differential EquationsSection 1:
Introduction
Section 2:
Laplace’s Equation; Steady-State Temperature in a Rectangular Plate
Section 3:
The Diffusion or Heat Flow Equation; The Schrodinger Equation
Section 4:
The wave Equation; The Vibrating String
Section 5:
Steady-State Temperature in a Cylinder
Section 6:
Vibration of a Circular Membrane
Section 7:
Steady-State Temperature in a Sphere
Section 8:
Poisson’s Equation
Section 9:
Integral Transform Solutions of Partial Differential Equations
Section 10:
Miscellaneous Problems
Chapter 14: Functions of a Complex VariableSection 1:
Introduction
Section 2:
Analytic Functions
Section 3:
Contour Integrals
Section 4:
Laurent Series
Section 5:
The Residue Theorem
Section 6:
Methods of Finding Residues
Section 7:
Evaluation of Definite Integrals by Use of the Residue Theorem
Section 8:
The Point at Infinity; Residues at Infinity
Section 9:
Mapping
Section 10:
Some Applications of Conformal Mapping
Section 11:
Miscellaneous Problems
Chapter 15: Probability and StatisticsSection 1:
Introduction
Section 2:
Sample Space
Section 3:
Probability Theorems
Section 4:
Methods of Counting
Section 5:
Random Variables
Section 6:
Continuous Distributions
Section 7:
Binomial Distribution
Section 8:
The Normal or Gaussian Distribution
Section 9:
The Poisson Distribution
Section 10:
Statistics and Experimental Measurements
Section 11:
Miscellaneous Problems
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