Advanced Calculus

by Fitzpatrick, Patrick M.

• ISBN13: 9780534376031
• ISBN10: 0534376037
• Edition: 2nd
• Format: Paperback
• Copyright: 2005-07-21
• Publisher: Brooks Cole

Summary

Easily master the fundamental concepts of mathematical analysis with ADVANCED CALCULUS. Presented in a clear and simple way, this advanced caluclus text leads you to a precise understanding of the subject by providing you with the tools you need to succeed. A wide variety of exercises helps you gain a genuine understanding of the material and examples demonstrate the significance of what you learn. Emphasizing the unity of the subject, the text shows that mathematical analysis is not a collection of isolated facts and techniques, but rather a coherent body of knowledge.

Table of Contents

Preface	xi
Preliminaries	1	(4)
Tools for Analysis	5	(18)
The Completeness Axiom and Some of Its Consequences	5	(7)
The Distribution of the Integers and the Rational Numbers	12	(4)
Inequalities and Identities	16	(7)
Convergent Sequences	23	(30)
The Convergence of Sequences	23	(12)
Sequences and Sets	35	(3)
The Monotone Convergence Theorem	38	(5)
The Sequential Compactness Theorem	43	(4)
Covering Properties of Sets	47	(6)
Continuous Functions	53	(34)
Continuity	53	(5)
The Extreme Value Theorem	58	(4)
The Intermediate Value Theorem	62	(4)
Uniform Continuity	66	(4)
The ε-δ Criterion for Continuity	70	(4)
Images and Inverses; Monotone Functions	74	(7)
Limits	81	(6)
Differentiation	87	(29)
The Algebra of Derivatives	87	(9)
Differentiating Inverses and Compositions	96	(5)
The Mean Value Theorem and Its Geometric Consequences	101	(10)
The Cauchy Mean Value Theorem and Its Analytic Consequences	111	(2)
The Notation of Leibnitz	113	(3)
Elementary Functions As Solutions of Differential Equations	116	(19)
Solutions of Differential Equations	116	(2)
The Natural Logarithm and Exponential Functions	118	(7)
The Trigonometric Functions	125	(7)
The Inverse Trigonometric Functions	132	(3)
Integration: Two Fundamental Theorems	135	(40)
Darboux Sums; Upper and Lower Integrals	135	(7)
The Archimedes--Riemann Theorem	142	(8)
Additivity, Monotonicity, and Linearity	150	(5)
Continuity and Integrability	155	(5)
The First Fundamental Theorem: Integrating Derivatives	160	(5)
The Second Fundamental Theorem: Differentiating Integrals	165	(10)
Integration: Further Topics	175	(24)
Solutions of Differential Equations	175	(3)
Integration by Parts and by Substitution	178	(5)
The Convergence of Darboux and Riemann Sums	183	(7)
The Approximation of Integrals	190	(9)
Approximation By Taylor Polynomials	199	(29)
Taylor Polynomials	199	(4)
The Lagrange Remainder Theorem	203	(6)
The Convergence of Taylor Polynomials	209	(3)
A Power Series for the Logarithm	212	(3)
The Cauchy Integral Remainder Theorem	215	(6)
A Nonanalytic, Infinitely Differentiable Function	221	(2)
The Weierstrass Approximation Theorem	223	(5)
Sequences and Series of Functions	228	(41)
Sequences and Series of Numbers	228	(13)
Pointwise Convergence of Sequences of Functions	241	(4)
Uniform Convergence of Sequences of Functions	245	(4)
The Uniform Limit of Functions	249	(6)
Power Series	255	(9)
A Continuous Nowhere Differentiable Function	264	(5)
The Euclidean Space Rn	269	(21)
The Linear Structure of Rn and the Scalar Product	269	(8)
Convergence of Sequences in Rn	277	(5)
Open Sets and Closed Sets in Rn	282	(8)
Continuity, Compactness, and Connectedness	290	(24)
Continuous Functions and Mappings	290	(8)
Sequential Compactness, Extreme Values, and Uniform Continuity	298	(6)
Pathwise Connectedness and the Intermediate Value Theorem	304	(6)
Connectedness and the Intermediate Value Property	310	(4)
Metric Spaces	314	(34)
Open Sets, Closed Sets, and Sequential Convergence	314	(8)
Completeness and the Contraction Mapping Principle	322	(6)
The Existence Theorem for Nonlinear Differential Equations	328	(9)
Continuous Mappings between Metric Spaces	337	(5)
Sequential Compactness and Connectedness	342	(6)
Differentiating Functions of Several Variables	348	(24)
Limits	348	(5)
Partial Derivatives	353	(11)
The Mean Value Theorem and Directional Derivatives	364	(8)
Local Approximation of Real-Valued Functions	372	(22)
First-Order Approximation, Tangent Planes, and Affine Functions	372	(8)
Quadratic Functions, Hessian Matrices, and Second Derivatives	380	(7)
Second-Order Approximation and the Second-Derivative Test	387	(7)
Approximating Nonlinear Mappings by Linear Mappings	394	(27)
Linear Mappings and Matrices	394	(13)
The Derivative Matrix and the Differential	407	(7)
The Chain Rule	414	(7)
Images and Inverses: The Inverse Function Theorem	421	(19)
Functions of a Single Variable and Maps in the Plane	421	(8)
Stability of Nonlinear Mappings	429	(4)
A Minimization Principle and the General Inverse Function Theorem	433	(7)
The Implicit Function Theorem and Its Applications	440	(30)
A Scalar Equation in Two Unknowns: Dini's Theorem	440	(9)
The General Implicit Function Theorem	449	(5)
Equations of Surfaces and Paths in R3	454	(6)
Constrained Extrema Problems and Lagrange Multipliers	460	(10)
Integrating Functions of Several Variables	470	(28)
Integration of Functions on Generalized Rectangles	470	(12)
Continuity and Integrability	482	(7)
Integration of Functions on Jordan Domains	489	(9)
Iterated Integration and Changes of Variables	498	(22)
Fubini's Theorem	498	(7)
The Change of Variables Theorem: Statements and Examples	505	(5)
Proof of the Change of Variables Theorem	510	(10)
Line and Surface Integrals	520	(39)
Arclength and Line Integrals	520	(13)
Surface Area and Surface Integrals	533	(10)
The Integral Formulas of Green and Stokes	543	(16)
A CONSEQUENCES OF THE FIELD AND POSITIVITY AXIOMS	559	(6)
A.1 The Field Axioms and Their Consequences	559	(4)
A.2 The Positivity Axioms and Their Consequences	563	(2)
B LINEAR ALGEBRA	565	(16)
Index	581