Table of Contents

Part 1 Linear Analysis

1 Linear Spaces
 1.1 Introduction
 1.2 The Definition of a Linear Space
 1.3 Examples of Linear Spaces
 1.4 Elementary Consequences of the Axioms
 1.5 Exercises
 1.6 Subspaces of a Linear Space
 1.7 Dependent and Independent Sets in a Linear Space
 1.8 Bases and Dimension
 1.9 Components
 1.10 Exercises
 1.11 Inner Products, Euclidean Spaces. Norms
 1.12 Orthogonality in a Euclidean Space
 1.13 Exercises
 1.14 Construction of Orthogonal Sets. The GramSchmidt Process
 1.15 Orthogonal Complements. Projections
 1.16 Best Approximation of Elements in a Euclidean Space by Elements in a FiniteDimensional Subspace

2 Linear Transformations and Matrices
 2.1 Linear Transformations
 2.2 Null Space and Range
 2.3 Nullity and Rank
 2.4 Exercises
 2.5 Algebraic Operations on Linear Transformations
 2.6 Inverses
 2.7 OnetoOne Linear Transformations
 2.8 Exercises
 2.9 Linear Transformations with Prescribed Values
 2.10 Matrix Representations of Linear Transformations
 2.11 Construction of a Matrix Representation in Diagonal Form
 2.12 Exercises
 2.13 Linear Spaces of Matrices
 2.14 Tsomorphism between Linear Transformations and Matrices
 2.15 Multiplication of Matrices
 2.16 Exercises
 2.17 Systems of Linear Equations
 2.18 Computation Techniques
 2.19 Inverses of Square Matrices
 2.20 Exercises
 2.21 Miscellaneous Exercises on Matrices

3 Determinants
 3.1 Introduction
 3.2 Motivation for the Choice of Axioms for a Determinant Function
 3.3 A Set of Axioms for a Determinant Function
 3.4 Computation of Determinants
 3.5 The Uniqueness Theorem
 3.6 Exercises
 3.7 The Product Formula for Determinants
 3.8 The Determinant of the Inverse of a Nonsingular Matrix
 3.9 Determinants and Independence of Vectors
 3.10 The Determinant of a BlockDiagonal Matrix
 3.11 Exercises
 3.12 Expansion Formulas for Determinants. Minors and Cofactors
 3.13 Existence of the Determinant Function
 3.14 The Determinant of a Transpose
 3.15 The Cofactor Matrix
 3.16 Cramer’s Rule

4 Eigenvalues and Eigenvectors
 4.1 Linear Transformations with Diagonal Matrix Representations
 4.2 Eigenvectors and Eigenvalues of a Linear Transformation
 4.3 Linear Independence of Eigenvectors Corresponding to Distinct Eigenvalues
 4.4 Exercises
 4.5 The FiniteDimensional Case. Characteristic Polynomials
 4.6 Calculation of Eigenvalues and Eigenvectors in the FiniteDimensional Case
 4.7 Trace of a Matrix
 4.8 Exercises
 4.9 Matrices Representing the Same Linear Transformation. Similar Matrices

5 Eigenvalues of Operators Acting on Euclidean Spaces
 5.1 Eigenvalues and Inner Products
 5.2 Hermitian and SkewHermitian Transformations
 5.3 Eigenvalues and Eigenvectors of Hermitian and SkewHermitian Operators
 5.4 Orthogonality of Eigenvectors Corresponding to Distinct Eigenvalues
 5.5 Exercises
 5.6 Existence of an Orthonormal Set of Eigenvectors for Hermitian and SkewHermitian Operators Acting on FiniteDimensional Spaces
 5.7 Matrix Representations for Hermitian and SkewHermitian Operators
 5.8 Hermitian and SkewHermitian Matrices. The Adjoint of a Matrix
 5.9 Diagonalization of a Hermitian or SkewHermitian Matrix
 5.10 Unitary Matrices. Orthogonal Matrices
 5.11 Exercises
 5.12 Quadratic Forms
 5.13 Reduction of a Real Quadratic Form to a Diagonal Form
 5.14 Applications to Analytic Geometry
 5.15 Exercises
 5.16 Eigenvalues of a Symmetric Transformation Obtained as Values of Its Quadratic Form
 5.17 Extremal Properties of Eigenvalues of a Symmetric Transformation
 5.18 The FiniteDimensional Case
 5.19 Unitary Transformations

6 Linear Differential Equations
 6.1 Historical Introduction
 6.2 Review of Results Concerning Linear Equations of First and Second Orders
 6.3 Exercises
 6.4 Linear Differential Equations of Order 𝑛
 6.5 The ExistenceUniqueness Theorem
 6.6 The Dimension of the Solution Space of a Homogeneous Linear Equation
 6.7 The Algebra of ConstantCoefficient Operators
 6.8 Determination of a Basis of Solutions for Linear Equations with Constant Coefficients by Factorization of Operators
 6.9 Exercises
 6.10 The Relation between the Homogeneous and Nonhomogeneous Equations
 6.11 Determination of a Particular Solution of the Nonhomogeneous Equation. The Method of Variation of Parameters
 6.12 Nonsingularity of the Wronskian Matrix of 𝑛 Independent Solutions of a Homogeneous Linear Equation
 6.13 Special Methods for Determining a Particular Solution of the Nonhomogeneous Equation. Reduction to a System of FirstOrder Linear Equations
 6.14 The Annihilator Method for Determining a Particular Solution of the Nonhomogeneous Equation
 6.15 Exercises
 6.16 Miscellaneous Exercises on Linear Differential Equations
 6.17 Linear Equations of Second Order with Analytic Coefficients
 6.18 The Legendre Equation
 6.19 The Legendre Polynomials
 6.20 Rodrigues’ Formula for the Legendre Polynomials
 6.21 Exercises
 6.22 The Method of Frobenius
 6.23 The Bessel Equation

7 Systems of Differential Equations
 7.1 Introduction
 7.2 Calculus of Matrix Functions
 7.3 Infinite Series of Matrices. Norms of Matrices
 7.4 Exercises
 7.5 The Exponential Matrix
 7.6 The Differential Equation Satisfied by 𝑒^𝑡𝐴
 7.7 Uniqueness Theorem for the Matrix Differential Equation 𝐹′(𝑡) = 𝐴𝐹(𝑡)
 7.8 The Law of Exponents for Exponential Matrices
 7.9 Existence and Uniqueness Theorems for Homogeneous Linear Systems with Constant Coefficients
 7.10 The Problem of Calculating 𝑒^𝑡𝐴
 7.11 The CayleyHamilton Theorem
 7.12 Exercises
 7.13 Putzer’s Method for Calculating 𝑒^𝑡𝐴
 7.14 Alternate Methods for Calculating 𝑒^𝑡𝐴 in Special Cases
 7.15 Exercises
 7.16 Nonhomogeneous Linear Systems with Constant Coefficients
 7.17 Exercises
 7.18 The General Linear System 𝑌′(𝑡) = 𝑃(𝑡)𝑌(𝑡) + 𝑄(𝑡)
 7.19 A PowerSeries Method for Solving Homogeneous Linear Systems
 7.20 Exercises
 7.21 Proof of the Existence Theorem by the Method of Successive Approximations
 7.22 The Method of Successive Approximations Applied to FirstOrder Nonlinear Systems
 7.23 Proof of an ExistenceUniqueness Theorem for FirstOrder Nonlinear Systems
 7.24 Exercises
 7.25 Successive Approximations and Fixed Points of Operators
 7.26 Normed Linear Spaces
 7.27 Contraction Operators
 7.28 FixedPoint Theorem for Contraction Operators
 7.29 Applications of the FixedPoint Theorem

1 Linear Spaces

Part 2 Nonlinear Analysis

8 Differential Calculus of Scalar and Vector Fields
 8.1 Functions from 𝐑ⁿ to R′′. Scalar and Vector Fields
 8.2 Open Balls and Open Sets
 8.3 Exercises
 8.4 Limits and Continuity
 8.5 Exercises
 8.6 The Derivative of a Scalar Field with Respect to a Vector
 8.7 Directional Derivatives and Partial Derivatives
 8.8 Partial Derivatives of Higher Order
 8.9 Exercises
 8.10 Directional Derivatives and Continuity
 8.11 The Total Derivative
 8.12 The Gradient of a Scalar Field
 8.13 A Sufficient Condition for Differentiability
 8.14 Exercises
 8.15 A Chain Rule for Derivatives of Scalar Fields
 8.16 Applications to Geometry. Level Sets. Tangent Planes
 8.17 Exercises
 8.18 Derivatives of Vector Fields
 8.19 Differentiability Implies Continuity
 8.20 The Chain Rule for Derivatives of Vector Fields
 8.21 Matrix Form of the Chain Rule
 8.22 Exercises
 8.23 Sufficient Conditions for the Equality of Mixed Partial Derivatives

9 Applications of the Differential Calculus
 9.1 Partial Differential Equations
 9.2 A FirstOrder Partial Differential Equation with Constant Coefficients
 9.3 Exercises
 9.4 The OneDimensional Wave Equation
 9.5 Exercises
 9.6 Derivatives of Functions Defined Implicitly
 9.7 Worked Examples
 9.8 Exercises
 9.9 Maxima, Minima, and Saddle Points
 9.10 SecondOrder Taylor Formula for Scalar Fields
 9.11 The Nature of a Stationary Point Determined by the Eigenvalues of the Hessian Matrix
 9.12 SecondDerivative Test for Extrema of Functions of Two Variables
 9.13 Exercises
 9.14 Extrema with Constraints. Lagrange’s Multipliers
 9.15 Exercises
 9.16 The ExtremeValue Theorem for Continuous Scalar Fields
 9.17 The SmallSpan Theorem for Continuous Scalar Fields (Uniform Continuity)

10 Line Integrals
 10.1 Introduction
 10.2 Paths and Line Integrals
 10.3 Other Notations for Line Integrals
 10.4 Basic Properties of Line Integrals
 10.5 Exercises
 10.6 The Concept of Work as a Line Integral
 10.7 Line Integrals with Respect to Arc Length
 10.8 Further Applications of Line Integrals
 10.9 Exercises
 10.10 Open Connected Sets. Independence of the Path
 10.11 The Second Fundamental Theorem of Calculus for Line Integrals
 10.12 Applications to Mechanics
 10.13 Exercises
 10.14 The First Fundamental Theorem of Calculus for Line Integrals
 10.15 Necessary and Sufficient Conditions for a Vector Field to Be a Gradient
 10.16 Necessary Conditions for a Vector Field to Be a Gradient
 10.17 Special Methods for Constructing Potential Functions
 10.18 Exercises
 10.19 Applications to Exact Differential Equations of First Order
 10.20 Exercises
 10.21 Potential Functions on Convex Sets

11 Multiple Integrals
 11.1 Introduction
 11.2 Partitions of Rectangles. Step Functions
 11.3 The Double Integral of a Step Function
 11.4 The Definition of the Double Integral of a Function Defined and Bounded on a Rectangle
 11.5 Upper and Lower Double Integrals
 11.6 Evaluation of a Double Integral by Repeated OneDimensional Integration
 11.7 Geometric Interpretation of the Double Integral as a Volume
 11.8 Worked Examples
 11.9 Exercises
 11.10 Integrability of Continuous Functions
 11.11 Integrability of Bounded Functions with Discontinuities
 11.12 Double Integrals Extended over More General Regions
 11.13 Applications to Area and Volume
 11.14 Worked Examples
 11.15 Exercises
 11.16 Further Applications of Double Integrals
 11.17 Two Theorems of Pappus
 11.18 Exercises
 11.19 Green’s Theorem in the Plane
 11.20 Some Applications of Green’s Theorem
 11.21 A Necessary and Sufficient Condition for a TwoDimensional Vector Field to Be a Gradient
 11.22 Exercises
 11.23 Green’s Theorem for Multiply Connected Regions
 11.24 The Winding Number
 11.25 Exercises
 11.26 Change of Variables in a Double Integral
 11.27 Special Cases of the Transformation Formula
 11.28 Exercises
 11.29 Proof of the Transformation Formula in a Special Case
 11.30 Proof of the Transformation Formula in the General Case
 11.31 Extensions to Higher Dimensions
 11.32 Change of Variables in an nFold Integral
 11.33 Worked Examples

12 Surface Integrals
 12.1 Parametric Representation of a Surface
 12.2 The Fundamental Vector Product
 12.3 The Fundamental Vector Product as a Normal to the Surface
 12.4 Exercises
 12.5 Area of a Parametric Surface
 12.6 Exercises
 12.7 Surface Integrals
 12.8 Change of Parametric Representation
 12.9 Other Notations for Surface Integrals
 12.10 Exercises
 12.11 The Theorem of Stokes
 12.12 The Curl and Divergence of a Vector Field
 12.13 Exercises
 12.14 Further Properties of the Curl and Divergence
 12.15 Exercises
 12.16 Reconstruction of a Vector Field from Its Curl
 12.17 Exercises
 12.18 Extensions of Stokes’ Theorem
 12.19 The Divergence Theorem (Gauss’ Theorem)
 12.20 Applications of the Divergence Theorem

8 Differential Calculus of Scalar and Vector Fields

Part 3 Special Topics

13 Set Functions and Elementary Probability
 13.1 Historical Introduction
 13.2 Finitely Additive Set Functions
 13.3 Finitely Additive Measures
 13.4 Exercises
 13.5 The Definition of Probability for Finite Sample Spaces
 13.6 Special Terminology Peculiar to Probability Theory
 13.7 Exercises
 13.8 Worked Examples
 13.9 Exercises
 13.10 Some Basic Principles of Combinatorial Analysis
 13.11 Exercises
 13.12 Conditional Probability
 13.13 Independence
 13.14 Exercises
 13.15 Compound Experiments
 13.16 Bernoulli Trials
 13.17 The Most Probable Number of Successes in 𝑛 Bernoulli Trials
 13.18 Exercises
 13.19 Countable and Uncountable Sets
 13.20 Exercises
 13.21 The Definition of Probability for Countably Infinite Sample Spaces
 13.22 Exercises
 13.23 Miscellaneous Exercises on Probability

14 Calculus of Probabilities
 14.1 The Definition of Probability for Uncountable Sample Spaces
 14.2 Countability of the Set of Points with Positive Probability
 14.3 Random Variables
 14.4 Exercises
 14.5 Distribution Functions
 14.6 Discontinuities of Distribution Functions
 14.7 Discrete Distributions. Probability Mass Functions
 14.8 Exercises
 14.9 Continuous Distributions. Density Functions
 14.10 Uniform Distribution over an Interval
 14.11 Cauchy’s Distribution
 14.12 Exercises
 14.13 Exponential Distributions
 14.14 Normal Distributions
 14.15 Remarks on More General Distributions
 14.16 Exercises
 14.17 Distributions of Functions of Random Variables
 14.18 Exercises
 14.19 Distributions of TwoDimensional Random Variables
 14.20 TwoDimensional Discrete Distributions
 14.21 TwoDimensional Continuous Distributions. Density Functions
 14.22 Exercises
 14.23 Distributions of Functions of Two Random Variables
 14.24 Exercises
 14.25 Expectation and Variance
 14.26 Expectation of a Function of a Random Variable
 14.27 Exercises
 14.28 Chebyshev’s Inequality
 14.29 Laws of Large Numbers
 14.30 The Central Limit Theorem of the Calculus of Probabilities

15 Introduction to Numerical Analysis
 15.1 Historical Introduction
 15.2 Approximations by Polynomials
 15.3 Polynomial Approximation and Normed Linear Spaces
 15.4 Fundamental Problems in Polynomial Approximation
 15.5 Exercises
 15.6 Interpolating Polynomials
 15.7 Equally Spaced Interpolation Points
 15.8 Error Analysis in Polynomial Interpolation
 15.9 Exercises
 15.10 Newton’s Interpolation Formula
 15.11 Equally Spaced Interpolation Points. The Forward Difference Operator
 15.12 Factorial Polynomials
 15.13 Exercises
 15.14 A Minimum Problem Relative to the Max Norm
 15.15 Chebyshev Polynomials
 15.16 A Minimal Property of Chebyshev Polynomials
 15.17 Application to the Error Formula for Interpolation
 15.18 Exercises
 15.19 Approximate Integration. The Trapezoidal Rule
 15.20 Simpson’s Rule
 15.21 Exercises
 15.22 The Euler Summation Formula

13 Set Functions and Elementary Probability