Calculus (Book)

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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
1. THE CONCEPTS OF INTEGRAL CALCULUS
1.1 The basic ideas of Cartesian geometry 48
1.2 Functions. Informal description and examples 50
1.3 Functions. Formal definition as a set of ordered pairs 53
1.4 More examples of real functions 54
1.5 Exercises 56
1.6 The concept of area as a set function 57
1.7 Exercises 60
1.8 Intervals and ordinate sets 60
1.9 Partitions and step functions 61
1.10 Sum and product of step functions 63
1.11 Exercises 63
1.12 The definition of the integral for step functions 64
1.13 Properties of the integral of a step function 66
1.14 Other notations for integrals 69
1.15 Exercises 70
1.16 The integral of more general functions 72
1.17 Upper and lower integrals 74
1.18 The area of an ordinate set expressed as an integral 75
1.19 Informal remarks on the theory and technique of integration 75
1.20 Monotonic and piecewise monotonic functions. Definitions and examples 76
1.21 Integrability of bounded monotonic functions 77
1.22 Calculation of the integral of a bounded monotonic function 79
1.23 Calculation of the integral \(\int_0^b x^p dx\) when \(p\) is a positive integer 79
1.24 The basic properties of the integral 80
1.25 Integration of polynomials 81
1.26 Exercises 83
1.27 Proofs of the basic properties of the integral 84
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