ADVANCED CALCULUS : theory and practice.

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
1 Real Numbers
	1.1 Axiomatic Systems
	1.2 Axioms of the Set R
	1.3 Some Consequences of the Completeness Axiom
	1.4 Some Thoughts about R
2 Sequences and Their Limits
	2.1 Computing the Limits: Review
	2.2 Definition of the Limit
	2.3 Properties of Limits
	2.4 Monotone Sequences
	2.5 Number e
	2.6 Cauchy Sequences
	2.7 Bolzano–Weierstrass Theorem
	2.8 Limit Superior and Limit Inferior
	2.9 Some Interesting Limits
3 Continuity
	3.1 Computing Limits of Functions: Review
	3.2 Review of Functions
	3.3 Continuous Functions: A Geometric Viewpoint
	3.4 Limits of Functions
	3.5 Other Limits
		3.5.1 One-Sided Limits
		3.5.2 Limits at Infinity
		3.5.3 Infinite Limits
	3.6 Properties of Continuous Functions
	3.7 Continuity of Elementary Functions
	3.8 Uniform Continuity
	3.9 Two Properties of Continuous Functions
4 Derivative
	4.1 Computing the Derivatives: A Review
	4.2 Derivative
	4.3 Rules of Differentiation
	4.4 Monotonicity: Local Extrema
	4.5 Taylor’s Formula
	4.6 L’Hôpital’s Rule
5 Indefinite Integral
	5.1 Computing Indefinite Integrals: A Review
	5.2 Antiderivative
		5.2.1 Rational Functions
		5.2.2 Irrational Functions
		5.2.3 Binomial Differentials
		5.2.4 Some Trigonometric Integrals
6 Definite Integral
	6.1 Computing Definite Integrals: A Review
	6.2 Definite Integral
	6.3 Integrable Functions
	6.4 Riemann Sums
	6.5 Properties of Definite Integrals
	6.6 Fundamental Theorem of Calculus
	6.7 Infinite and Improper Integrals
		6.7.1 Infinite Integrals
		6.7.2 Improper Integrals
7 Infinite Series
	7.1 Review of Infinite Series
	7.2 Definition of a Series
	7.3 Series with Positive Terms
	7.4 Root and Ratio Tests
		7.4.1 Additional Tests for Convergence
	7.5 Series with Arbitrary Terms
		7.5.1 Additional Tests for Convergence
		7.5.2 Rearrangement of a Series
8 Sequences and Series of Functions
	8.1 Convergence of a Sequence of Functions
	8.2 Uniformly Convergent Sequences of Functions
	8.3 Function Series
		8.3.1 Applications to Differential Equations
		8.3.2 Continuous Nowhere Differentiable Function
	8.4 Power Series
	8.5 Power Series Expansions of Elementary Functions
9 Fourier Series
	9.1 Introduction
	9.2 Pointwise Convergence of Fourier Series
	9.3 Uniform Convergence of Fourier Series
	9.4 Cesàro Summability
	9.5 Mean Square Convergence of Fourier Series
	9.6 Influence of Fourier Series
10 Functions of Several Variables
	10.1 Subsets of Rn
	10.2 Functions and Their Limits
	10.3 Continuous Functions
	10.4 Boundedness of Continuous Functions
	10.5 Open Sets in Rn
	10.6 Intermediate Value Theorem
	10.7 Compact Sets
11 Derivatives of Functions of Several Variables
	11.1 Computing Derivatives: Review
	11.2 Derivatives and Differentiability
	11.3 Properties of the Derivative
	11.4 Functions from Rn to Rm
	11.5 Taylor’s Formula
	11.6 Extreme Values
12 Implicit Functions and Optimization
	12.1 Implicit Functions
	12.2 Derivative as a Linear Map
	12.3 Open Mapping Theorem
	12.4 Implicit Function Theorem
	12.5 Constrained Optimization
	12.6 The Second Derivative Test for Constrained Optimization
		12.6.1 Absolute Extrema
13 Integrals Depending on a Parameter
	13.1 Uniform Convergence
	13.2 Integral as a Function
	13.3 Uniform Convergence of Improper Integrals
	13.4 Improper Integral as a Function
	13.5 Some Important Integrals
14 Integration in Rn
	14.1 Double Integrals over Rectangles
	14.2 Double Integrals over Jordan Sets
	14.3 Double Integrals as Iterated Integrals
	14.4 Transformations of Jordan Sets in R2
	14.5 Change of Variables in Double Integrals
	14.6 Improper Integrals
	14.7 Multiple Integrals
15 Fundamental Theorems of Multivariable Calculus
	15.1 Curves in Rn
	15.2 Line Integrals
	15.3 Green’s Theorem
	15.4 Surface Integrals
	15.5 Divergence Theorem
	15.6 Stokes’ Theorem
	15.7 Differential Forms on Rn
	15.8 Exact Differential Forms on Rn
16 Solutions and Answers to Selected Exercises
Bibliography
Subject Index
Author Index