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Advanced Calculus (Second Edition) by Patrick M. Fitzpatrick



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Advanced Calculus (Second Edition) by Patrick M. Fitzpatrick , University of Maryland, College Park. Advanced Calculus is intended as a text for courses that furnish the backbone of the student's undergraduate education in mathematical analysis. The goal is to rigorously present the fundamental concepts within the context of illuminating examples and stimulating exercises. This book is self-contained and starts with the creation of basic tools using the completeness axiom. The continuity, differentiability, integrability, and power series representation properties of functions of a single variable are established. The next few chapters describe the topological and metric properties of Euclidean space. These are the basis of a rigorous treatment of differential calculus (including the Implicit Function Theorem and Lagrange Multipliers) for mappings between Euclidean spaces and integration for functions of several real variables. Special attention has been paid to the motivation for proofs. Selected topics, such as the Picard Existence Theorem for differential equations, have been included in such a way that selections may be made while preserving a fluid presentation of the essential material. Supplemented with numerous exercises, Advanced Calculus is a perfect book for undergraduate students of analysis.

Advanced Calculus (Second Edition) by Patrick M. Fitzpatrick cover the following topics.



  • Preface
    Preliminaries

  • 1. TOOLS FOR ANALYSIS
    1.1 The Completeness Axiom and Some of Its Consequences
    1.2 The Distribution of the Integers and the Rational Numbers
    1.3 Inequalities and Identities

  • 2. CONVERGENT SEQUENCES
    2.1 The Convergence of Sequences
    2.2 Sequences and Sets
    2.3 The Monotone Convergence Theorem
    2.4 The Sequential Compactness Theorem
    2.5 Covering Properties of Sets*

  • 3. CONTINUOUS FUNCTIONS
    3.1 Continuity
    3.2 The Extreme Value Theorem
    3.3 The Intermediate Value Theorem
    3.4 Uniform Continuity
    3.5 The E-o Criterion for Continuity
    3.6 Images and Inverses; Monotone Functions
    3.7 Limits

  • 4. DIFFERENTIATION
    4.1 The Algebra of Derivatives
    4.2 Differentiating Inverses and Compositions
    4.3 The Mean Value Theorem and Its Geometric Consequences
    4.4 The Cauchy Mean Value Theorem and Its Analytic Consequences
    4.5 The Notation of Leibnitz

  • 5. ELEMENTARY FUNCTIONS AS SOLUTIONS OF DIFFERENTIAL EQUATIONS
    5.1 Solutions of Differential Equations
    5.2 The Natural Logarithm and Exponential Functions
    5.3 The Trigonometric Functions
    5.4 The Inverse Trigonometric Functions

  • 6. INTEGRATION: TWO FUNDAMENTAL THEOREMS
    6.1 Darboux Sums; Upper and Lower Integrals
    6.2 The Archimedes-Riemann Theorem
    6.3 Additivity, Monotonicity, and Linearity
    6.4 Continuity and Integrability
    6.5 The First Fundamental Theorem: Integrating Derivatives
    6.6 The Second Fundamental Theorem: Differentiating Integrals

  • 7.* INTEGRATION: FURTHER TOPICS
    7.1 Solutions of Differential Equations
    7.2 Integration by Parts and by Substitution
    7.3 The Convergence of Darboux and Riemann Sums
    7.4 The Approximation of Integrals

  • 8. APPROXIMATION BY TAYLOR POLYNOMIALS
    8.1 Taylor Polynomials
    8.2 The Lagrange Remainder Theorem
    8.3 The Convergence of Taylor Polynomials
    8.4 A Power Series for the Logarithm
    8.5 The Cauchy Integral Remainder Theorem
    8.6 A Nonanalytic, Infinitely Differentiable Function
    8.7 The Weierstrass Approximation Theorem

  • 9. SEQUENCES AND SERIES OF FUNCTIONS
    9.1 Sequences and Series of Numbers
    9.2 Pointwise Convergence of Sequences of Functions
    9.3 Uniform Convergence of Sequences of Functions
    9.4 The Uniform Limit of Functions
    9.5 Power Series
    9.6 A Continuous Nowhere Differentiable Function

  • 10. THE EUCLIDEAN SPACE IR.n
    10.1 The Linear Structure of IR.n and the Scalar Product
    10.2 Convergence of Sequences in IR.n
    10.3 Open Sets and Closed Sets in IR.n

  • 11. CONTINUITY, COMPACTNESS, AND CONNECTEDNESS
    11.1 Continuous Functions and Mappings
    11.2 Sequential Compactness, Extreme Values, and Uniform Continuity
    11.3 Pathwise Connectedness and the Intermediate Value Theorem*
    11.4 Connectedness and the Intermediate Value Property*
    12.* METRIC SPACES
    12.1 Open Sets, Closed Sets, and Sequential Convergence
    12.2 Completeness and the Contraction Mapping Principle
    12.3 The Existence Theorem for Nonlinear Differential Equations
    12.4 Continuous Mappings between Metric Spaces
    12.5 Sequential Compactness and Connectedness

  • 13. DIFFERENTIATING FUNCTIONS OF SEVERAL VARIABLES
    13.1 Limits
    13.2 Partial Derivatives
    13.3 The Mean Value Theorem and Directional Derivatives

  • 14. LOCAL APPROXIMATION OF REAL-VALUED FUNCTIONS
    14.1 First-Order Approximation, Tangent Planes, and Affine Functions
    14.2 Quadratic Functions, Hessian Matrices, and Second Derivatives*
    14.3 Second-Order Approximation and the Second-Derivative Test*

  • 15. APPROXIMATING NONLINEAR MAPPINGS BY LINEAR MAPPINGS
    15.1 Linear Mappings and Matrices
    15.2 The Derivative Matrix and the Differential
    15.3 The Chain Rule

  • 16. IMAGES AND INVERSES: THE INVERSE FUNCTION THEOREM
    16.1 Functions of a Single Variable and Maps in the Plane
    16.2 Stability of Nonlinear Mappings
    16.3 A Minimization Principle and the General Inverse Function Theorem

  • 17. THE IMPLICIT FUNCTION THEOREM AND ITS APPLICATIONS
    17.1 A Scalar Equation in Two Unknowns: Dini
    s Theorem
    17.2 The General Implicit Function Theorem
    17.3 Equations of Surfaces and Paths in IR3
    17.4 Constrained Extrema Problems and Lagrange Multipliers

  • 18. INTEGRATING FUNCTIONS OF SEVERAl VARIABLES
    18.1 Integration of Functions on Generalized Rectangles
    18.2 Continuity and Integrability
    18.3 Integration of Functions on jordan Domains

  • 19. ITERATED INTEGRATION AND CHANGES OF VARIABLES
    19.1 Fubini
    s Theorem
    19.2 The Change of Variables Theorem: Statements and Examples
    19.3 Proof of the Change of Variables Theorem

  • 20. LINE AND SURFACE INTEGRALS
    20.1 Arclength and Line Integrals
    20.2 Surface Area and Surface Integrals
    20.3 The Integral Formulas of Green and Stokes

  • Appendix
    A. CONSEQUENCES OF THE FIELD AND POSITIVITY AXIOMS
    A.1 The Field Axioms and Their Consequences
    A.2 The Positivity Axioms and Their Consequences
    B.LINEAR ALGEBRA
    Index

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