Editing Calculus (Book)

{{Stub}}
{{InfoboxBook
|title=Calculus
|image=[[File:Apostol Calculus V1 Cover.jpg]]
|author=[https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol]
|language=English
|series=
|genre=
|publisher=Wiley
|publicationdate=16 January 1991
|pages=666
|isbn10=0471000051
|isbn13=978-0471000051
}}
The textbook '''''Calculus''''' by [https://en.wikipedia.org/wiki/Tom_M._Apostol Tom Apostol] introduces calculus. 

== Table of Contents ==

{| class="wikitable"
|-
! Chapter/Section # !! Title !! Page #
|- 
! colspan="3" | I. INTRODUCTION
|-
! colspan="3" | 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
|-
| <nowiki>*</nowiki>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
|- 
! colspan="3" | 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
|-
! colspan="3" | Part 3: A set of Axioms for the Real-Number System
|-
| I 3.1 || Introduction || 17
|-
| I 3.2 || The field axioms || 17
|-
| <nowiki>*</nowiki>I 3.3 || Exercises || 19
|-
| I 3.4 || The order axioms || 19
|-
| <nowiki>*</nowiki>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
|-
| <nowiki>*</nowiki>I 3.12 || Exercises || 28
|-
| <nowiki>*</nowiki>I 3.13 || Existence of square roots of nonnegative real numbers || 29
|-
| <nowiki>*</nowiki>I 3.14 || Roots of higher order. Rational powers || 30
|-
| <nowiki>*</nowiki>I 3.15 || Representation of real numbers by decimals || 30
|-
! colspan="3" | 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
|-
| <nowiki>*</nowiki>I 4.3 || The well-ordering principle || 34
|-
| I 4.4 || Exercises || 35
|-
| <nowiki>*</nowiki>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
|-
| <nowiki>*</nowiki>I 4.10 || Miscellaneous exercises involving induction || 44
|-
! colspan="3" | 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
|-
! colspan="3" | 2. SOME APPLICATIONS OF INTEGRATION
|-
| 2.1 || Introduction || 88
|-
| 2.2 || The area of a region between two graphs expressed as an integral || 88
|-
| 2.3 || Worked examples || 89
|-
| 2.4 || Exercises || 94
|-
| 2.5 || The trigonometric functions || 94
|-
| 2.6 || Integration formulas for the sine and cosine || 94
|-
| 2.7 || A geometric description of the sine and cosine functions || 94
|-
| 2.8 || Exercises || 94
|-
| 2.9 || Polar coordinates || 94
|-
| 2.10 || The integral for area in polar coordinates || 94
|-
| 2.11 || Exercises || 94
|-
| 2.12 || Application of integration to the calculation of volume || 94
|-
| 2.13 || Exercises || 94
|-
| 2.14 || Application of integration to the calculation of work || 94
|-
| 2.15 || Exercises || 94
|-
| 2.16 || Average value of a function || 94
|-
| 2.17 || Exercises || 94
|-
| 2.18 || The integral as a function of the upper limit. Indefinite integrals || 94
|-
| 2.19 || Exercises || 94
|-
! colspan="3" | 3. CONTINUOUS FUNCTIONS
|-
| 3.1 || Informal description of continuity || 126
|-
| 3.2 || The definition of the limit of a function || 127
|-
| 3.3 || The definition of continuity of a function || 130
|-
| 3.4 || The basic limit theorems. More examples of continuous functions || 131
|-
| 3.5 || Proofs of the basic limit theorems || 135
|-
| 3.6 || Exercises || 138
|-
| 3.7 || Composite functions and continuity || 140
|-
| 3.8 || Exercises || 142
|-
| 3.9 || Bolzano's theorem for continuous functions || 142
|-
| 3.10 || The intermediate-value theorem for continuous functions || 144
|-
| 3.11 || Exercises || 145
|-
| 3.12 || The process of inversion || 146
|-
| 3.13 || Properties of functions preserved by inversion || 147
|-
| 3.14 || Inverses of piecewise monotonic functions || 148
|-
| 3.15 || Exercises || 149
|-
| 3.16 || The extreme-value theorem for continuous functions || 150
|-
| 3.17 || The small-span theorem for continuous functions (uniform continuity) || 152
|-
| 3.18 || The integrability theorem for continuous functions || 152
|-
| 3.19 || Mean-value theorems for integrals of continuous functions || 154
|-
| 3.20 || Exercises || 155
|-
! colspan="3" | 4. DIFFERENTIAL CALCULUS
|-
| 4.1 || Historical introduction || 156
|-
| 4.2 || A problem involving velocity || 157
|-
| 4.3 || The derivative of a function || 159
|-
| 4.4 || Examples of derivatives || 161
|-
| 4.5 || The algebra of derivatives || 164
|-
| 4.6 || Exercises || 167
|-
| 4.7 || Geometric interpretation of the derivative as a slope || 169
|-
| 4.8 || Other notations for derivatives || 171
|-
| 4.9 || Exercises || 173
|-
| 4.10 || The chain rule for differentiating composite functions || 174
|-
| 4.11 || Applications of the chain rule. Related rates and implicit differentiation || 176
|-
| 4.12 || Exercises || 179
|-
| 4.13 || Applications of the differentiation to extreme values of cuntions|| 181
|-
| 4.14 || The mean-value theorem for derivatives || 183
|-
| 4.15 || Exercises || 186
|-
| 4.16 || Applications of the mean-value theorem to geometric properties of functions || 187
|-
| 4.17 || Second-derivative test for extrema || 188
|-
| 4.18 || Curve sketching || 189
|-
| 4.19 || Exercises || 191
|-
| 4.20 || Worked examples of extremum problems || 191
|-
| 4.21 || Exercises || 194
|-
| 4.22 || Partial derivatives || 196
|-
| 4.23 || Exercises || 201
|-
! colspan="3" | 5. THE RELATION BETWEEN INTEGRATION AND DIFFERENTIATION
|-
| 1 || Coordinate systems || 191
|-
| 2 || Distance between points || 197
|-
| 3 || Equation of a circle || 203
|-
| 4 || Rational points on a circle || 206
|-
! colspan="3" | Chapter 9: Operations on Points
|-
| 1 || Dilations and reflections || 213
|-
| 2 || Addition, subtraction, and the parallelogram law || 218
|-
! colspan="3" | Chapter 10: Segments, Rays, and Lines
|-
| 1 || Segments || 229
|-
| 2 || Rays || 231
|-
| 3 || Lines || 236
|-
| 4 || Ordinary equation for a line || 246
|-
! colspan="3" | 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
|-
! colspan="3" | 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
|-
! colspan="3" | PART IV: MISCELLANEOUS
|-
! colspan="3" | 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
|-
! colspan="3" | Chapter 14: Mappings
|-
| 1 || Definition || 345
|-
| 2 || Formalism of mappings || 351
|-
| 3 || Permutations || 359
|-
! colspan="3" | Chapter 15: Complex Numbers
|-
| 1 || The complex plane || 375
|-
| 2 || Polar form || 380
|-
! colspan="3" | Chapter 16: Induction and Summations
|-
| 1 || Induction || 383
|-
| 2 || Summations || 388
|-
| 3 || Geometric series || 396
|-
! colspan="3" | 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
|-
! colspan="2" | Index || 429
|-
|}

[[Category:Mathematics]]