Advanced Calculus

Contents......Page f005.djvu
Preface......Page f011.djvu
About the Author......Page f018.djvu
Preliminaries......Page p001.djvu
1.1 The Completeness Axiom and Some of Its Consequences......Page p005.djvu
1.2 The Distribution of the Integers and the Rational Numbers......Page p012.djvu
1.3 Inequalities and Identities......Page p016.djvu
2.1 The Convergence of Sequences......Page p023.djvu
2.2 Sequences and Sets......Page p035.djvu
2.3 The Monotone Convergence Theorem......Page p038.djvu
2.4 The Sequential Compactness Theorem......Page p043.djvu
2.5 Covering Properties of Sets*......Page p047.djvu
3.1 Continuity......Page p053.djvu
3.2 The Extreme Value Theorem......Page p058.djvu
3.3 The Intermediate Value Theorem......Page p062.djvu
3.4 Uniform Continuity......Page p066.djvu
3.5 The epsilon-delta Criterion for Continuity......Page p070.djvu
3.6 Images and Inverses; Monotone Functions......Page p074.djvu
3.7 Limits......Page p081.djvu
4.1 The Algebra of Derivatives......Page p087.djvu
4.2 Differentiating Inverses and Compositions......Page p096.djvu
4.3 The Mean Value Theorem and Its Geometric Consequences......Page p101.djvu
4.4 The Cauchy Mean Value Theorem and Its Analytic Consequences......Page p111.djvu
4.5 The Notation of Leibnitz......Page p113.djvu
5.1 Solutions of Differential Equations......Page p116.djvu
5.2 The Natural Logarithm and Exponential Functions......Page p118.djvu
5.3 The Trigonometric Functions......Page p125.djvu
5.4 The Inverse Trigonometric Functions......Page p132.djvu
6.1 Darboux Sums; Upper and Lower Integrals......Page p135.djvu
6.2 The Archimedes-Riemann Theorem......Page p142.djvu
6.3 Additivity, Monotonicity, and Linearity......Page p150.djvu
6.4 Continuity and Integrability......Page p155.djvu
6.5 The First Fundamental Theorem: Integrating Derivatives......Page p160.djvu
6.6 The Second Fundamental Theorem: Differentiating Integrals......Page p165.djvu
7.1 Solutions of Differential Equations......Page p175.djvu
7.2 Integration by Parts and by Substitution......Page p178.djvu
7.3 The Convergence of Darboux and Riemann Sums......Page p183.djvu
7.4 The Approximation of Integrals......Page p190.djvu
8.1 Taylor Polynomials......Page p199.djvu
8.2 The Lagrange Remainder Theorem......Page p203.djvu
8.3 The Convergence of Taylor Polynomials......Page p209.djvu
8.4 A Power Series for the Logarithm......Page p212.djvu
8.5 The Cauchy Integral Remainder Theorem......Page p215.djvu
8.6 A Nonanalytic, Infinitely Differentiable Function......Page p221.djvu
8.7 The Weierstrass Approximation Theorem......Page p223.djvu
9.1 Sequences and Series of Numbers......Page p228.djvu
9.2 Pointwise Convergence of Sequences of Functions......Page p241.djvu
9.3 Uniform Convergence of Sequences of Functions......Page p245.djvu
9.4 The Uniform Limit of Functions......Page p249.djvu
9.5 Power Series......Page p255.djvu
9.6 A Continuous Nowhere Differentiable Function......Page p264.djvu
10.1 The Linear Structure of R^n and the Scalar Product......Page p269.djvu
10.2 Convergence of Sequences in R^n......Page p277.djvu
10.3 Open Sets and Closed Sets in R^n......Page p282.djvu
11.1 Continuous Functions and Mappings......Page p290.djvu
11.2 Sequential Compactness, Extreme Values, and Uniform Continuity......Page p298.djvu
11.3 Pathwise Connectedness and the Intermediate Value Theorem*......Page p304.djvu
11.4 Connectedness and the Intermediate Value Property*......Page p310.djvu
12.1 Open Sets, Closed Sets, and Sequential Convergence......Page p314.djvu
12.2 Completeness and the Contraction Mapping Principle......Page p322.djvu
12.3 The Existence Theorem for Nonlinear Differential Equations......Page p328.djvu
12.4 Continuous Mappings between Metric Spaces......Page p337.djvu
12.5 Sequential Compactness and Connectedness......Page p342.djvu
13.1 Limits......Page p348.djvu
13.2 Partial Derivatives......Page p353.djvu
13.3 The Mean Value Theorem and Directional Derivatives......Page p364.djvu
14.1 First-Order Approximation, Tangent Planes, and Affine Functions......Page p372.djvu
14.2 Quadratic Functions, Hessian Matrices, and Second Derivatives*......Page p380.djvu
14.3 Second-Order Approximation and the Second-Derivative Test*......Page p387.djvu
15.1 Linear Mappings and Matrices......Page p394.djvu
15.2 The Derivative Matrix and the Differential......Page p407.djvu
15.3 The Chain Rule......Page p414.djvu
16.1 Functions of a Single Variable and Maps in the Plane......Page p421.djvu
16.2 Stability of Nonlinear Mappings......Page p429.djvu
16.3 A Minimization Principle and the General Inverse Function Theorem......Page p433.djvu
17.1 A Scalar Equation in Two Unknowns: Dini\'s Theorem......Page p440.djvu
17.2 The General Implicit Function Theorem......Page p449.djvu
17.3 Equations of Surfaces and Paths in R^3......Page p454.djvu
17.4 Constrained Extrema Problems and Lagrange Multipliers......Page p460.djvu
18.1 Integration of Functions on Generalized Rectangles......Page p470.djvu
18.2 Continuity and Integrability......Page p482.djvu
18.3 Integration of Functions on Jordan Domains......Page p489.djvu
19.1 Fubini\'s Theorem......Page p498.djvu
19.2 The Change of Variables Theorem: Statements and Examples......Page p505.djvu
19.3 Proof of the Change of Variables Theorem......Page p510.djvu
20.1 Arclength and Line Integrals......Page p520.djvu
20.2 Surface Area and Surface Integrals......Page p533.djvu
20.3 The Integral Formulas of Green and Stokes......Page p543.djvu
A.1 The Field Axioms and Their Consequences......Page p559.djvu
A.2 The Positivity Axioms and Their Consequences......Page p563.djvu
B Linear Algebra......Page p565.djvu
Index......Page p581.djvu