Calculus (Book)

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Calculus
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