Alignment: Calculus • Volume 1 • One-Variable Calculus, with an Introduction to Linear Algebra • Second Edition

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Table of Contents

  • 1 Introduction
    • Part 1 Historical Introduction
      • 1.1 The Two Basic Concepts of Calculus
      • 1.2 Historical Background
      • 1.3 The Method of Exhaustion for the Area of a Parabolic Segment
      • 1.4 Exercises
      • 1.5 A Critical Analysis of Archimedes’ Method
      • 1.6 The Approach to Calculus to Be Used in This Book
    • Part 2 Some Basic Concepts of the Theory of Sets
      • 2.1 Introduction to Set Theory
      • 2.2 Notations for Designating Sets
      • 2.3 Subsets
      • 2.4 Unions, Intersections, Complements
      • 2.5 Exercises
    • Part 3 A Set of Axioms for the Real-Number System
      • 3.1 Introduction
      • 3.2 The Field Axioms
      • 3.3 Exercises
      • 3.4 The Order Axioms
      • 3.5 Exercises
      • 3.6 Integers and Rational Numbers
      • 3.7 Geometric Interpretation of Real Numbers as Points on a Line
      • 3.8 Upper Bound of a Set, Maximum Element, Least Upper Bound (Supremum)
      • 3.9 The Least-Upper-Bound Axiom (Completeness Axiom)
      • 3.10 The Archimedean Property of the Real-Number System
      • 3.11 Fundamental Properties of the Supremum and Infimum
      • 3.12 Exercises
      • 3.13 Existence of Square Roots of Nonnegative Real Numbers
      • 3.14 Roots of Higher Order. Rational Powers
      • 3.15 Representation of Real Numbers by Decimals
    • Part 4 Mathematical Induction, Summation Notation, and Related Topics
      • 4.1 An Example of a Proof by Mathematical Induction
      • 4.2 The Principle of Mathematical Induction
      • 4.3 The Well-Ordering Principle
      • 4.4 Exercises
      • 4.5 Proof of the Well-Ordering Principle
      • 4.6 The Summation Notation
      • 4.7 Exercises
      • 4.8 Absolute Values and the Triangle Inequality
      • 4.9 Exercises
      • 4.10 Miscellaneous Exercises Involving Induction
  • 1 The Concepts of Integral Calculus
    • 1.1 The Basic Ideas of Cartesian Geometry
    • 1.2 Functions. Informal Description and Examples
    • 1.3 Functions. Formal Definition as a Set of Ordered Pairs
    • 1.4 More Examples of Real Functions
    • 1.5 Exercises
    • 1.6 The Concept of Area as a Set Function
    • 1.7 Exercises
    • 1.8 Intervals and Ordinate Sets
    • 1.9 Partitions and Step Functions
    • 1.10 Sum and Product of Step Functions
    • 1.11 Exercises
    • 1.12 The Definition of the Integral for Step Functions
    • 1.13 Properties of the Integral of a Step Function
    • 1.14 Other Notations for Integrals
    • 1.15 Exercises
    • 1.16 The Integral of More General Functions
    • 1.17 Upper and Lower Integrals
    • 1.18 The Area of an Ordinate Set Expressed as an Integral
    • 1.19 Informal Remarks on the Theory and Technique of Integration
    • 1.20 Monotonic and Piecewise Monotonic Functions. Definitions and Examples
    • 1.21 Integrability of Bounded Monotonic Functions
    • 1.22 Calculation of the Integral of a Bounded Monotonic Function
    • 1.23 Calculation of the Integral ∫^𝑏_0 𝑥^𝑝 𝑑𝑥 When 𝑝 Is a Positive Integer
    • 1.24 The Basic Properties of the Integral
    • 1.25 Integration of Polynomials
    • 1.26 Exercises
    • 1.27 Proofs of the Basic Properties of the Integral
  • 2 Some Applications of Integration
  • 3 Continuous Functions
    • 3.1 Informal Description of Continuity
    • 3.2 The Definition of the Limit of a Function
    • 3.3 The Definition of Continuity of a Function
    • 3.4 The Basic Limit Theorems. More Examples of Continuous Functions
    • 3.5 Proofs of the Basic Limit Theorems
    • 3.6 Exercises
    • 3.7 Composite Functions and Continuity
    • 3.8 Exercises
    • 3.9 Bolzano’s Theorem for Continuous Functions
    • 3.10 The Intermediate-Value Theorem for Continuous Functions
    • 3.11 Exercises
    • 3.12 The Process of Inversion
    • 3.13 Properties of Functions Preserved by Inversion
    • 3.14 Inverses of Piecewise Monotonic Functions
    • 3.15 Exercises
    • 3.16 The Extreme-Value Theorem for Continuous Functions
    • 3.17 The Small-Span Theorem for Continuous Functions (Uniform Continuity)
    • 3.18 The Integrability Theorem for Continuous Functions
    • 3.19 Mean-Value Theorems for Integrals of Continuous Functions
    • 3.20 Exercises
  • 4 Differential Calculus
  • 5 The Relation between Integration and Differentiation
  • 6 The Logarithm, the Exponential, and the Inverse Trigonometric Functions
  • 7 Polynomial Approximations to Functions
    • 7.1 Introduction
    • 7.2 The Taylor Polynomials Generated by a Function
    • 7.3 Calculus of Taylor Polynomials
    • 7.4 Exercises
    • 7.5 Taylor’s Formula with Remainder
    • 7.6 Estimates for the Error in Taylor’s Formula
    • 7.7 Other Forms of the Remainder in Taylor’s Formula
    • 7.8 Exercises
    • 7.9 Further Remarks on the Error in Taylor’s Formula. The o-Notation
    • 7.10 Applications to Indeterminate Forms
    • 7.11 Exercises
    • 7.12 L’Hôpital’s Rule for the Indeterminate Form 0/0
    • 7.13 Exercises
    • 7.14 The Symbols +∞ and −∞. Extension of L’Hôpital’s Rule
    • 7.15 Infinite Limits
    • 7.16 The Behavior of Log 𝑥 and 𝑒^𝑥 for Large 𝑥
    • 7.17 Exercises
  • 8 Introduction to Differential Equations
    • 8.1 Introduction
    • 8.2 Terminology and Notation
    • 8.3 A First-Order Differential Equation for the Exponential Function
    • 8.4 First-Order Linear Differential Equations
    • 8.5 Exercises
    • 8.6 Some Physical Problems Leading to First-Order Linear Differential Equations
    • 8.7 Exercises
    • 8.8 Linear Equations of Second Order with Constant Coefficients
    • 8.9 Existence of Solutions of the Equation 𝑦′′ + 𝑏𝑦 = 0
    • 8.10 Reduction of the General Equation to the Special Case 𝑦′′ + 𝑏𝑦 = 0
    • 8.11 Uniqueness Theorem for the Equation 𝑦′′ + 𝑏𝑦 = 0
    • 8.12 Complete Solution of the Equation 𝑦′′ + 𝑏𝑦 = 0
    • 8.13 Complete Solution of the Equation 𝑦′′ + 𝑎𝑦′ + 𝑏𝑦 = 0
    • 8.14 Exercises
    • 8.15 Nonhomogeneous Linear Equations of Second Order with Constant Coefficients
    • 8.16 Special Methods for Determining a Particular Solution of the Nonhomogeneous Equation 𝑦′′ + 𝑎𝑦′ + 𝑏𝑦 = 𝑅
    • 8.17 Exercises
    • 8.18 Examples of Physical Problems Leading to Linear Second-Order Equations with Constant Coefficients
    • 8.19 Exercises
    • 8.20 Remarks Concerning Nonlinear Differential Equations
    • 8.21 Integral Curves and Direction Fields
    • 8.22 Exercises
    • 8.23 First-Order Separable Equations
    • 8.24 Exercises
    • 8.25 Homogeneous First-Order Equations
    • 8.26 Exercises
    • 8.27 Some Geometrical and Physical Problems Leading to First-Order Equations
    • 8.28 Miscellaneous Review Exercises
  • 9 Complex Numbers
  • 10 Sequences, Infinite Series, Improper Integrals
  • 11 Sequences and Series of Functions
    • 11.1 Pointwise Convergence of Sequences of Functions
    • 11.2 Uniform Convergence of Sequences of Functions
    • 11.3 Uniform Convergence and Continuity
    • 11.4 Uniform Convergence and Integration
    • 11.5 A Sufficient Condition for Uniform Convergence
    • 11.6 Power Series. Circle of Convergence
    • 11.7 Exercises
    • 11.8 Properties of Functions Represented by Real Power Series
    • 11.9 The Taylor’s Series Generated by a Function
    • 11.10 A Sufficient Condition for Convergence of a Taylor’s Series
    • 11.11 Power-Series Expansions for the Exponential and Trigonometric Functions
    • 11.12 Bernstein’s Theorem
    • 11.13 Exercises
    • 11.14 Power Series and Differential Equations
    • 11.15 The Binomial Series
    • 11.16 Exercises
  • 12 Vector Algebra
  • 13 Applications of Vector Algebra to Analytic Geometry
    • 13.1 Introduction
    • 13.2 Lines in n-Space
    • 13.3 Some Simple Properties of Straight Lines
    • 13.4 Lines and Vector-Valued Functions
    • 13.5 Exercises
    • 13.6 Planes in Euclidean n-Space
    • 13.7 Planes and Vector-Valued Functions
    • 13.8 Exercises
    • 13.9 The Cross Product
    • 13.10 The Cross Product Expressed as a Determinant
    • 13.11 Exercises
    • 13.12 The Scalar Triple Product
    • 13.13 Cramer’s Rule for Solving a System of Three Linear Equations
    • 13.14 Exercises
    • 13.15 Normal Vectors to Planes
    • 13.16 Linear Cartesian Equations for Planes
    • 13.17 Exercises
    • 13.18 The Conic Sections
    • 13.19 Eccentricity of Conic Sections
    • 13.20 Polar Equations for Conic Sections
    • 13.21 Exercises
    • 13.22 Conic Sections Symmetric about the Origin
    • 13.23 Cartesian Equations for the Conic Sections
    • 13.24 Exercises
    • 13.25 Miscellaneous Exercises on Conic Sections
  • 14 Calculus of Vector-Valued Functions
    • 14.1 Vector-Valued Functions of a Real Variable
    • 14.2 Algebraic Operations. Components
    • 14.3 Limits, Derivatives, and Integrals
    • 14.4 Exercises
    • 14.5 Applications to Curves. Tangency
    • 14.6 Applications to Curvilinear Motion. Velocity, Speed, and Acceleration
    • 14.7 Exercises
    • 14.8 The Unit Tangent, the Principal Normal, and the Osculating Plane of a Curve
    • 14.9 Exercises
    • 14.10 The Definition of Arc Length
    • 14.11 Additivity of Arc Length
    • 14.12 The Arc-Length Function
    • 14.13 Exercises
    • 14.14 Curvature of a Curve
    • 14.15 Exercises
    • 14.16 Velocity and Acceleration in Polar Coordinates
    • 14.17 Plane Motion with Radial Acceleration
    • 14.18 Cylindrical Coordinates
    • 14.19 Exercises
    • 14.20 Applications to Planetary Motion
    • 14.21 Miscellaneous Review Exercises
  • 15 Linear Spaces
    • 15.1 Introduction
    • 15.2 The Definition of a Linear Space
    • 15.3 Examples of Linear Spaces
    • 15.4 Elementary Consequences the Axioms
    • 15.5 Exercises
    • 15.6 Subspaces of a Linear Space
    • 15.7 Dependent and Independent Sets in a Linear Space
    • 15.8 Bases and Dimension
    • 15.9 Exercises
    • 15.10 Inner Products, Euclidian Spaces, Norms
    • 15.11 Orthogonality in a Euclidean Space
    • 15.12 Exercises
    • 15.13 Construction of Orthogonal Sets. The Gram-Schmidt Process
    • 15.14 Orthogonal Complements. Projections
    • 15.15 Best Approximation of Elements in a Euclidean Space by Elements in a Finite-Dimensional Subspace
    • 15.16 Exercises
  • 16 Linear Transformations and Matrices
    • 16.1 Linear Transformations
    • 16.2 Nul1 Space and Range
    • 16.3 Nullity and Rank
    • 16.4 Exercises
    • 16.5 Algebraic Operations on Linear Transformations
    • 16.6 Inverses
    • 16.7 One-to-One Linear Transformations
    • 16.8 Exercises
    • 16.9 Linear Transformations with Prescribed Values
    • 16.10 Matrix Representations of Linear Transformations
    • 16.11 Construction of a Matrix Representation in Diagonal Form
    • 16.12 Exercises
    • 16.13 Linear Spaces of Matrices
    • 16.14 Isomorphism between Linear Transformations and Matrices
    • 16.15 Multiplication of Matrices
    • 16.16 Exercises
    • 16.17 Systems of Linear Equations
    • 16.18 Computation Techniques
    • 16.19 Inverses of Square Matrices
    • 16.20 Exercises
    • 16.21 Miscellaneous Exercises on Matrices

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