# Alignment: Calculus • Volume 2 • Multi Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability • Second Edition

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• 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 Gram-Schmidt Process
• 1.15 Orthogonal Complements. Projections
• 1.16 Best Approximation of Elements in a Euclidean Space by Elements in a Finite-Dimensional 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 One-to-One 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 Block-Diagonal 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 Finite-Dimensional Case. Characteristic Polynomials
• 4.6 Calculation of Eigenvalues and Eigenvectors in the Finite-Dimensional 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 Skew-Hermitian Transformations
• 5.3 Eigenvalues and Eigenvectors of Hermitian and Skew-Hermitian 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 Skew-Hermitian Operators Acting on Finite-Dimensional Spaces
• 5.7 Matrix Representations for Hermitian and Skew-Hermitian Operators
• 5.8 Hermitian and Skew-Hermitian Matrices. The Adjoint of a Matrix
• 5.9 Diagonalization of a Hermitian or Skew-Hermitian Matrix
• 5.10 Unitary Matrices. Orthogonal Matrices
• 5.11 Exercises
• 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 Finite-Dimensional 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 Existence-Uniqueness Theorem
• 6.6 The Dimension of the Solution Space of a Homogeneous Linear Equation
• 6.7 The Algebra of Constant-Coefficient 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 First-Order 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 Cayley-Hamilton 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 Power-Series 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 First-Order Nonlinear Systems
• 7.23 Proof of an Existence-Uniqueness Theorem for First-Order 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 Fixed-Point Theorem for Contraction Operators
• 7.29 Applications of the Fixed-Point Theorem
• 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 First-Order Partial Differential Equation with Constant Coefficients
• 9.3 Exercises
• 9.4 The One-Dimensional 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 Second-Order Taylor Formula for Scalar Fields
• 9.11 The Nature of a Stationary Point Determined by the Eigenvalues of the Hessian Matrix
• 9.12 Second-Derivative 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 Extreme-Value Theorem for Continuous Scalar Fields
• 9.17 The Small-Span 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 One-Dimensional 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 Two-Dimensional 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 n-Fold 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
• Part 3 Special Topics
• 13 Set Functions and Elementary Probability
• 13.1 Historical Introduction
• 13.2 Finitely Additive Set Functions
• 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 Two-Dimensional Random Variables
• 14.20 Two-Dimensional Discrete Distributions
• 14.21 Two-Dimensional 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