In a calculus course, the student is expected to acquire a great number of
techniques and problem-solving devices and to put them to practical use. The
goal of an advanced calculus course is to put the calculus material into proper
perspective and to ask the question, "Why does calculus work?" In an
overwhelming number of cases, the answer is simple: "Because the real numbers have
the least-upper-bound property." This text is what I hope to be an informative
and entertaining documentation of this answer. At the same time, I have tried
to prepare the reader for topics beyond advanced calculus such as topology,
theory of functions, real variables and measure theory, functional analysis,
integration on manifolds, and last but not least, what is generally referred to as
applied mathematics. To avoid misunderstandings, let it be stressed that this
text is not an introduction to any of these subjects but rather a preparation
for them.
Contents:
Preface vii
1 Numbers 1
2 Functions 58
3 The Derivative 120
4 The Riemann Integral 141
5 The Euclidean n-Space 197
6 Vector-Valued Functions of a Vector Variable 226
7 Sequences of Functions 265
8 Linear Functions 296
9 The Derivative of a Vector-Valued Function of a Vector Variable 312
10 Nonlinear Functions 359
11 Multiple Integrals 392
12 Transformation of Integrals 438
13 Line and Surface Integrals 470
14 Infinite Series 566
Appendix 1 629
Appendix 2 631
Index 663
Author(s): Hans Sagan
Publisher: Houghton Mifflin Company
Year: 1974
Language: English
Pages: 685
City: Boston
Preface......Page f007.djvu
Contents......Page f011.djvu
1.1 Set Language—A Mathematical Shorthand......Page p001.djvu
1.2 Generalized Unions and Intersections: DeMorgan's Law......Page p006.djvu
*1.3 The Russell Paradox......Page p007.djvu
1.4 Cartesian Products and Functions......Page p008.djvu
1.5 The Inverse Function......Page p014.djvu
1.6 The Field of Rational Numbers......Page p020.djvu
1.7 Ordered Fields......Page p023.djvu
1.8 The Real Numbers as a Complete Ordered Field......Page p028.djvu
1.9 Properties of the Least Upper Bound......Page p032.djvu
*1.10 Representation of the Real Numbers by Decimals......Page p036.djvu
*1.11 Existence and Uniqueness of a Complete Ordered Field......Page p041.djvu
1.12 The Hierarchy of Numbers......Page p046.djvu
1.13 Cardinal Numbers......Page p048.djvu
*1.14 The Cantor Set......Page p055.djvu
2.15 Limit of a Sequence......Page p058.djvu
2.16 Three Fundamental Properties of Convergent Sequences......Page p061.djvu
2.17 Manipulations of Sequences......Page p063.djvu
2.18 Monotonic Sequences......Page p068.djvu
2.19 The Geometric Series......Page p070.djvu
2.20 The Number Line......Page p073.djvu
2.21 The Nested-Interval Property of R......Page p075.djvu
2.22 The Location of the Rationals and Irrationals Relative to the Reals......Page p078.djvu
2.23 Accumulation Point......Page p080.djvu
2.24 Limit of a Function......Page p083.djvu
2.25 Limit of a Function—Neighborhood Definition......Page p087.djvu
2.26 Continuous Functions......Page p090.djvu
2.27 Properties of Continuous Functions......Page p095.djvu
2.28 Some Elementary Point Set Topology......Page p100.djvu
2.29 Unions and Intersections of Open and Closed Sets......Page p103.djvu
2.30 The Bolzano-Weierstrass Theorem......Page p107.djvu
2.31 The Heine-Borel Property......Page p108.djvu
2.32 Compact Sets......Page p110.djvu
2.33 Uniformly Continuous Functions......Page p113.djvu
2.34 The Continuous Image of a Compact Set......Page p116.djvu
3.35 Definition of the Derivative......Page p120.djvu
3.36 Differentiation of a Composite Function......Page p124.djvu
3.37 Mean-Value Theorems......Page p128.djvu
3.38 The Intermediate-Value Property of Derivatives......Page p136.djvu
*3.39 The Differential......Page p138.djvu
4.40 Area Measure......Page p141.djvu
4.41 Properties of Area Measure......Page p143.djvu
*4.42 Area Measure of Rectangles......Page p146.djvu
4.43 Approximation by Polygonal Regions......Page p148.djvu
4.44 Upper and Lower Sums......Page p151.djvu
4.45 The Riemann Integral......Page p154.djvu
4.46 The Riemann Criterion for Integrability......Page p157.djvu
4.47 Integration of Continuous Functions and Monotonic Functions......Page p161.djvu
4.48 Sets of Lebesgue Measure Zero......Page p165.djvu
4.49 Characterization of Integrable Functions......Page p170.djvu
4.50 The Linearity of the Integral......Page p176.djvu
4.51 Properties of the Riemann Integral......Page p178.djvu
4.52 The Riemann Integral with a Variable Upper Limit......Page p182.djvu
*4.53 Improper Integrals......Page p190.djvu
5.54 Introduction......Page p197.djvu
5.55 The Cartesian n-Space......Page p198.djvu
5.56 Dot Product......Page p201.djvu
5.57 Norm......Page p204.djvu
5.58 The Euclidean n-Space......Page p208.djvu
6.54 Open and Closed Sets in E^n......Page p211.djvu
5.54 The Nested-Interval Property in E^n and the Bolzano-Weierstrass Theorem......Page p215.djvu
5.55 Sequences in E^n......Page p218.djvu
5.56 Cauchy Sequences......Page p221.djvu
6.63 Notation......Page p226.djvu
6.64 Limit of a Function......Page p229.djvu
6.65 Continuity......Page p231.djvu
6.66 Direct and Inverse Images......Page p233.djvu
6.67 Composite Functions......Page p240.djvu
6.68 Global Characterization of Continuous Functions......Page p243.djvu
6.69 Open Maps......Page p248.djvu
6.70 Compact Sets and the Heine-Borel Theorem......Page p249.djvu
6.71 The Continuous Image of a Compact Set......Page p252.djvu
6.72 Connected Sets......Page p253.djvu
6.73 The Continuous Image of a Connected Set......Page p257.djvu
6.74 Uniformly Continuous Functions......Page p260.djvu
6.75 Contraction Mappings......Page p262.djvu
7.76 Pointwise Convergence......Page p265.djvu
7.77 Uniform Convergence......Page p271.djvu
7.78 Uniformly Convergent Sequences of Continuous Functions and of Bounded Functions......Page p275.djvu
*7.79 Weierstrass Approximation Theorem......Page p278.djvu
*7.80 A Continuous Function that is Nowhere Differentiable......Page p281.djvu
7.81 Termwise Integration of Sequences......Page p285.djvu
7.82 Termwise Differentiation of Sequences......Page p290.djvu
8.83 Definition and Representation......Page p296.djvu
8.84 Linear Onto Functions and Linear One-To-One Functions......Page p300.djvu
8.85 Properties of Linear Functions......Page p306.djvu
9.86 Definition of the Derivative......Page p312.djvu
9.87 The Directional Derivative......Page p316.djvu
9.88 Partial Derivatives......Page p321.djvu
9.89 Representation of the Derivative......Page p323.djvu
9.90 Existence of the Derivative......Page p327.djvu
9.91 Differentiation Rules......Page p334.djvu
9.92 Differentiation of Composite Functions......Page p338.djvu
9.93 Mean-Value Theorems......Page p344.djvu
9.94 Partial Derivatives of Higher Order......Page p346.djvu
*9.95 The Inverse-Function Theorem and the Implicit-Function Theorem......Page p352.djvu
10.96 One-To-One Functions......Page p359.djvu
10.97 Onto Functions......Page p366.djvu
10.98 The Inverse-Function Theorem......Page p371.djvu
10.99 The Implicit-Function Theorem......Page p375.djvu
10.100 Implicit Differentiation......Page p380.djvu
*10.101 Extreme Values......Page p384.djvu
11.102 The Riemann Integral in E^n......Page p392.djvu
11.103 Existence of the Integral......Page p396.djvu
11.104 The Riemann Integral over Point Sets Other Than Intervals......Page p399.djvu
11.105 Jordan Content......Page p404.djvu
11.106 Properties of Jordan-Measurable Sets......Page p409.djvu
11.107 Integrals over Jordan-Measurable Sets......Page p411.djvu
11.108 Integration by Iteration—Fubini's Theorem......Page p415.djvu
11.109 Applications of Fubini's Theorem......Page p420.djvu
*11.110 Differentiation of an Integral with Respect to a Parameter......Page p424.djvu
*11.111 Transformation of Double Integrals......Page p429.djvu
12.112 Images of Jordan Measurable Sets......Page p438.djvu
12.113 Jordan Content of Linear Images of Jordan Measurable Sets......Page p445.djvu
12.114 Jordan Content of General Images of Intervals......Page p450.djvu
12.115 Transformation of Multiple Integrals: Jacobi's Theorem......Page p457.djvu
*12.116 Jordan Content as Length, Area, and Volume Measure......Page p464.djvu
13.117 Introduction......Page p470.djvu
13.118 Curves......Page p473.djvu
13.119 Surfaces......Page p481.djvu
13.120 Smooth Manifolds......Page p485.djvu
13.121 Diffeomorphisms and Smooth Equivalence......Page p492.djvu
13.122 Tangent Lines and Tangent Vectors......Page p498.djvu
13.123 Tangent Planes and Normal Vectors......Page p503.djvu
13.124 Patches and Quilts......Page p510.djvu
13.125 Arc Length......Page p514.djvu
13.126 Surface Area......Page p520.djvu
13.127 Differential Forms......Page p524.djvu
13.128 Work and Steady Flow......Page p534.djvu
13.129 Differentials of k-Forms......Page p541.djvu
13.130 Green's Theorem......Page p544.djvu
13.131 Stokes'Theorem......Page p552.djvu
13.132 The Theorem of Gauss......Page p560.djvu
14.133 Convergence of Infinite Series......Page p566.djvu
14.134 The Integral Test......Page p570.djvu
14.135 Absolute Convergence and Conditional Convergence......Page p574.djvu
14.136 Tests for Absolute Convergence......Page p580.djvu
*14.137 The CBS Inequality and the Triangle Inequality for Infinite Series......Page p588.djvu
14.138 Double Series......Page p590.djvu
14.139 Cauchy Products......Page p595.djvu
14.140 Infinite Series of Functions......Page p598.djvu
14.141 Power Series......Page p603.djvu
*14.142 The Exponential Function and Its Inverse Function......Page p607.djvu
*14.143 The Trigonometric Functions and Their Inverse Functions......Page p612.djvu
14.144 Manipulations with Power Series......Page p618.djvu
14.145 Taylor Series......Page p622.djvu
Cast of Characters (in Order of Their Appearance)......Page p629.djvu
The Greek Alphabet......Page p630.djvu
Answers and Hints......Page p631.djvu
Index......Page p663.djvu
