Calculus (Book): Difference between revisions

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| <nowiki>*</nowiki>I 1.4 || Exercises || 8
| <nowiki>*</nowiki>I 1.4 || Exercises || 8
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| I 1.5 || Rational numbers || 8
| I 1.5 || A critical analysis of the Archimedes' method || 8
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| I 1.6 || Multiplicative inverses || 10
| I 1.6 || The approach to calculus to be used in this book || 10
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! colspan="3" | Chapter 2: Linear Equations
! colspan="3" | Part 2: Some Basic Concepts of the Theory of Sets
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| 1 || Equations in two unknowns || 53
| I 2.1 || Introduction to set theory || 11
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| 2 || Equations in three unknowns || 57
| I 2.2 || Notations for designating sets || 12
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! colspan="3" | Chapter 3: Real Numbers
| I 2.3 || Subsets || 12
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| 1 || Addition and multiplication || 61
| I 2.4 || Unions, intersections, complements || 13
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| 2 || Real numbers: positivity || 64
| I 2.5 || Exercises || 15
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| 3 || Powers and roots || 70
! colspan="3" | Part 3: A set of Axioms for the Real-Number System
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| 4 || Inequalities || 75
| I 3.1 || Introduction || 17
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! colspan="3" | Chapter 4: Quadratic Equations
| I 3.2 || The field axioms || 17
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| <nowiki>*</nowiki>I 3.3 || Exercises || 19
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| I 3.4 || The order axioms || 19
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| <nowiki>*</nowiki>I 3.5 || Exercises || 21
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| I 3.6 || Integers and rational numbers || 21
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| I 3.7 || Geometric interpretation of real numbers as points on a line || 22
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| I 3.8 || Upper bound of a set, maximum element, least upper bound (supremum) || 23
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| I 3.9 || The least-Upper-bound axiom (completeness axiom) || 25
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| I 3.10 || The Archimedean property of the real-number system || 25
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| I 3.11 || Fundamental properties of the supremum and infimum || 26
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| <nowiki>*</nowiki>I 3.12 || Exercises || 28
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| <nowiki>*</nowiki>I 3.13 || Existence of square roots of nonnegative real numbers || 29
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| <nowiki>*</nowiki>I 3.14 || Roots of higher order. Rational powers || 30
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| <nowiki>*</nowiki>I 3.15 || Representation of real numbers by decimals || 30
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! colspan="3" | Part 4: Mathematical Induction, Summation Notation, and Related Topics
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| I 4.1 || An example of a proof by mathematical induction || 32
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| I 4.2 || The principle of mathematical induction || 34
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| <nowiki>*</nowiki>I 4.3 || The well-ordering principle || 34
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| I 4.4 || Exercises || 35
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| <nowiki>*</nowiki>I 4.5 || Proof of the well-ordering principle || 37
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| I 4.6 || The summation notation || 37
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| I 4.7 || Exercises || 39
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| I 4.8 || Absolute values and the triangle inequality || 41
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| I 4.9 || Exercises || 43
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| <nowiki>*</nowiki>I 4.10 || Miscellaneous exercises involving induction || 44
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! colspan="3" | Interlude: On Logic and Mathematical Expressions
! colspan="3" | Interlude: On Logic and Mathematical Expressions

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Calculus
Apostol Calculus V1 Cover.jpg
Information
Author Tom Apostol
Language English
Publisher Wiley
Publication Date 16 January 1991
Pages 666
ISBN-10 0471000051
ISBN-13 978-0471000051

The textbook Calculus by Tom Apostol introduces calculus.

Table of Contents

Chapter/Section # Title Page #
I. INTRODUCTION
Part 1: Historical Introduction
I 1.1 The two basic concepts of calculus 1
I 1.2 Historical background 2
I 1.3 The method of exhaustion for the area of a parabolic segment 3
*I 1.4 Exercises 8
I 1.5 A critical analysis of the Archimedes' method 8
I 1.6 The approach to calculus to be used in this book 10
Part 2: Some Basic Concepts of the Theory of Sets
I 2.1 Introduction to set theory 11
I 2.2 Notations for designating sets 12
I 2.3 Subsets 12
I 2.4 Unions, intersections, complements 13
I 2.5 Exercises 15
Part 3: A set of Axioms for the Real-Number System
I 3.1 Introduction 17
I 3.2 The field axioms 17
*I 3.3 Exercises 19
I 3.4 The order axioms 19
*I 3.5 Exercises 21
I 3.6 Integers and rational numbers 21
I 3.7 Geometric interpretation of real numbers as points on a line 22
I 3.8 Upper bound of a set, maximum element, least upper bound (supremum) 23
I 3.9 The least-Upper-bound axiom (completeness axiom) 25
I 3.10 The Archimedean property of the real-number system 25
I 3.11 Fundamental properties of the supremum and infimum 26
*I 3.12 Exercises 28
*I 3.13 Existence of square roots of nonnegative real numbers 29
*I 3.14 Roots of higher order. Rational powers 30
*I 3.15 Representation of real numbers by decimals 30
Part 4: Mathematical Induction, Summation Notation, and Related Topics
I 4.1 An example of a proof by mathematical induction 32
I 4.2 The principle of mathematical induction 34
*I 4.3 The well-ordering principle 34
I 4.4 Exercises 35
*I 4.5 Proof of the well-ordering principle 37
I 4.6 The summation notation 37
I 4.7 Exercises 39
I 4.8 Absolute values and the triangle inequality 41
I 4.9 Exercises 43
*I 4.10 Miscellaneous exercises involving induction 44
Interlude: On Logic and Mathematical Expressions
1 On reading books 93
2 Logic 94
3 Sets and elements 99
4 Notation 100
PART II: INTUITIVE GEOMETRY
Chapter 5: Distance and Angles
1 Distance 107
2 Angles 110
3 The Pythagoras theorem 120
Chapter 6: Isometries
1 Some standard mappings of the plane 133
2 Isometries 143
3 Composition of isometries 150
4 Inverse of isometries 155
5 Characterization of isometries 163
6 Congruences 166
Chapter 7: Area and Applications
1 Area of a disc of radius r 173
2 Circumference of a circle of radius r 180
PART III: COORDINATE GEOMETRY
Chapter 8: Coordinates and Geometry
1 Coordinate systems 191
2 Distance between points 197
3 Equation of a circle 203
4 Rational points on a circle 206
Chapter 9: Operations on Points
1 Dilations and reflections 213
2 Addition, subtraction, and the parallelogram law 218
Chapter 10: Segments, Rays, and Lines
1 Segments 229
2 Rays 231
3 Lines 236
4 Ordinary equation for a line 246
Chapter 11: Trigonometry
1 Radian measure 249
2 Sine and cosine 252
3 The graphs 264
4 The tangent 266
5 Addition formulas 272
6 Rotations 277
Chapter 12: Some Analytic Geometry
1 The straight line again 281
2 The parabola 291
3 The ellipse 297
4 The hyperbola 300
5 Rotation of hyperbolas 305
PART IV: MISCELLANEOUS
Chapter 13: Functions
1 Definition of a function 313
2 Polynomial functions 318
3 Graphs of functions 330
4 Exponential function 333
5 Logarithms 338
Chapter 14: Mappings
1 Definition 345
2 Formalism of mappings 351
3 Permutations 359
Chapter 15: Complex Numbers
1 The complex plane 375
2 Polar form 380
Chapter 16: Induction and Summations
1 Induction 383
2 Summations 388
3 Geometric series 396
Chapter 17: Determinants
1 Matrices 401
2 Determinants of order 2 406
3 Properties of 2 x 2 determinants 409
4 Determinants of order 3 414
5 Properties of 3 x 3 determinants 418
6 Cramer's Rule 424
Index 429
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