Advanced Calculus (Pure and Applied Undergraduate Texts

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...

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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.

Preface
Tools for Analysis
The Completeness Axiom and Some of Its Consequences
The Distribution of the Integers and the Rational Numbers
Inequalities and Identities
Convergent Sequences
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Preface
Tools for Analysis
The Completeness Axiom and Some of Its Consequences
The Distribution of the Integers and the Rational Numbers
Inequalities and Identities
Convergent Sequences
The Convergence of Sequences
Sequences and Sets
The Monotone Convergence Theorem
The Sequential Compactness Theorem
Covering Properties of Sets
Continuous Functions
Continuity
The Extreme Value Theorem
The Intermediate Value Theorem
Uniform Continuity
The Criterion for Continuity
Images and Inverses; Monotone Functions
Limits
Differentiation
The Algebra of Derivatives
Differentiating Inverses and Compositions
The Mean Value Theorem and Its Geometric Consequences
The Cauchy Mean Value Theorem and Its Analytic Consequences
The Notation of Leibnitz
Elementary Functions as Solutions of Differential Equations
Solutions of Differential Equations
The Natural Logarithm and Exponential Functions
The Trigonometric Functions
The Inverse Trigonometric Functions
Integration: Two Fundamental Theorems
Darboux Sums; Upper and Lower Integrals
The Archimedes-Riemann Theorem
Additivity, Monotonicity, and Linearity
Continuity and Integrability
The First Fundamental Theorem: Integrating Derivatives
The Second Fundamental Theorem: Differentiating Integrals
Integration: Further Topics
Solutions of Differential Equations
Integration by Parts and by Substitution
The Convergence of Darboux and Riemann Sums
The Approximation of Integrals
Approximation By Taylor Polynomials
Taylor Polynomials
The Lagrange Remainder Theorem
The Convergence of Taylor Polynomials
A Power Series for the Logarithm
The Cauchy Integral Remainder Theorem
A Nonanalytic, Infinitely Differentiable Function
The Weierstrass Approximation Theorem
Sequences And Series Of Functions
Sequences and Series of Numbers
Pointwise Convergence of Sequences of Functions
Uniform Convergence of Sequences of Functions
The Uniform Limit of Functions
Power Series
A Continuous Nowhere Differentiable Function
The Euclidean Space
The Linear Structure of and the Scalar Product
Convergence of Sequences in Rn
Open Sets and Closed Sets in Rn
Continuity, Compactness, And Connectedness
Continuous Functions and Mappings
Sequential Compactness, Extreme Values, and Uniform Continuity
Pathwise Connectedness and the Intermediate Value Theorem
Connectedness and the Intermediate Value Property
Metric Spaces
Open Sets, Closed Sets, and Sequential Convergence
Completeness and the Contraction Mapping Principle
The Existence Theorem for Nonlinear Differential Equations
Continuous Mappings between Metric Spaces
Sequential Compactness and Connectedness
Differentiating Functions Of Several Variables
Limits
Partial Derivatives
The Mean Value Theorem and Directional Derivatives
Local Approximation Of Real-Valued Functions
First-Order Approximation, Tangent Planes, and Affine Functions
Quadratic Functions, Hessian Matrices, and Second Derivatives
Second-Order Approximation and the Second-Derivative Test
Approximating Nonlinear Mappings By Linear Mappings
Linear Mappings and Matrices
The Derivative Matrix and the Differential
The Chain Rule
Images And Inverses: The Inverse Function Theorem
Functions of a Single Variable and Maps in the Plane
Stability of Nonlinear Mappings
A Minimization Principle and the General Inverse Function Theorem
The Implicit Function Theorem And Its Applications
A Scalar Equation in Two Unknowns: Dini's Theorem
The General Implicit Function Theorem
Equations of Surfaces and Paths in
Constrained Extrema Problems and Lagrange Multipliers
Integrating Functions Of Several Variables
Integration of Functions on Generalized Rectangles
Continuity and Integrability
Integration of Functions on Jordan Domains
Iterated Integration And Changes Of Variables
Fubini's Theorem
The Change of Variables Theorem: Statements and Examples
Proof of the Change of Variables Theorem
Line And Surface Integrals
Arclength and Line Integrals
Surface Area and Surface Integrals
The Integral Formulas of Green and Stokes
Appendix A Consequences Of The Field And Positivity Axioms
Appendix B Linear Algebra
Index
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