The Fundamentals of Mathematical Analysis

The Fundamentals of Mathematical Analysis, Volume I is a translation from Russian and used as a textbook that provides a systematic and serious discussion of the fundamentals of mathematical analysis. Differential and integral calculus, which were established in the 17th century; the theory of limits as the foundation of mathematical analysis; and the concept of real numbers as justifying the theory of limits are discussed in the early part of the volume. The concept of the limit of a function related to the concept of continuity of a function as explained by Bolzano and Cauchy is discussed. Fermat's theorem and Taylor's formula in understanding differential calculus are then presented. The book also investigates the value of the function when several independent variables occur, by considering the values of all the independent variables simultaneously. The concept of a primitive function and of an indefinite integral used in science and engineering is noted. Integral calculus, as found in geometric and mechanical applications, and differential calculus, as significant in some geometric application, are then examined. In the last part, the text discusses the works of Newton and Leibniz. The book will serve as a guide and reference for mathematicians, students, and professors of calculus and advanced mathematics.

Foreword

Chapter 1 Real Numbers

§ 1. The Set of Real Numbers and Its Ordering

1. Introductory Remarks

2. Definition of Irrational Number

3. Ordering of the Set of Real Numbers

4. Representation of a Real Number by an Infinite Decimal Fraction

5. Continuity of the Set of Real Numbers

6. Bounds of Number Sets

§ 2. Arithmetical Operations over Real Numbers

7. Definition and Properties of a Sum of Real Numbers

8. Symmetric Numbers. Absolute Quantity

9. Definition and Properties of a Product of Real Numbers

§ 3. Further Properties and Applications of Real Numbers

10. Existence of a Root. Power with a Rational Exponent

11. Power with an Arbitrary Real Exponent

12. Logarithms

13. Measuring Segments

Chapter 2 Functions of One Variable

§ 1. The Concept of a Function

14. Variable Quantity

15. The Domain of Variation of a Variable Quantity

16. Functional Relation between Variables. Examples

17. Definition of the Concept of Function

18. Analytic Method of Prescribing a Function

19. Graph of a Function

20. Functions of Positive Integral Argument

21. Historical Remarks

§ 2. Important Classes of Functions

22. Elementary Functions

23. The Concept of the Inverse Function

24. Inverse Trigonometric Functions

25. Superposition of Functions. Concluding Remarks

Chapter 3 Theory of Limits

§ 1. The Limit of a Function

26. Historical Remarks

27. Numerical Sequence

28. Definition of the Limit of a Sequence

29. Infinitesimal Quantities

30. Examples

31. Infinitely Large Quantities

32. Definition of the Limit of a Function

33. Another Definition of the Limit of a Function

34. Examples

35. One-Sided Limits

§ 2. Theorems on Limits

36. Properties of Functions of a Positive Integral Argument, Possessing a Finite Limit

37. Extension to the Case of a Function of a n Arbitrary Variable

38. Passage to the Limit in Equalities and Inequalities

39. Theorems on Infinitesimals

40. Arithmetical Operations on Variables

41. Indefinite Expressions

42. Extension to the Case of a Function of an Arbitrary Variable

43. Examples

§ 3. Monotonic Functions

44. Limit of a Monotonic Function of a Positive Integral Argument

45. Examples

46. a Lemma on Imbedded Intervals

47. The Limit of a Monotonic Function in the General Case

§ 4. The Number E

48. The Number E Defined as the Limit of a Sequence

49. Approximate Computation of the Number E

50. The Basic Formula for the Number E. Natural Logarithms

§ 5. The Principle of Convergence

51. Partial Sequences

52. The Condition of Existence of a Finite Limit for a Function of Positive Integral Argument

53. The Condition of Existence of a Finite Limit for a Function of an Arbitrary Argument

§ 6. Classification of Infinitely Small and Infinitely Large Quantities

54. Comparison of Infinitesimals

55. The Scale of Infinitesimals

56. Equivalent Infinitesimals

57. Separation of the Principal Part

58. Problems

59. Classification of Infinitely Large Quantities

Chapter 4 Continuous Functions of One Variable

§ 1. Continuity (and Discontinuity) of a Function

60. Definition of the Continuity of a Function at a Point

61. Condition of Continuity of a Monotonic Function

62. Arithmetical Operations over Continuous Functions

63. Continuity of Elementary Functions

64. The Superposition of Continuous Functions

65. Computation of Certain Limits

66. Power-Exponential Expressions

67. Classification of Discontinuities. Examples

§ 2. Properties of Continuous Functions

68. Theorem on the Zeros of a Function

69. Application to the Solution of Equations

70. Mean Value Theorem

71. The Existence of Inverse Functions

72. Theorem on The Boundedness of a Function

73. The Greatest and Smallest Values of a Function

74. The Concept of Uniform Continuity

75. Theorem on Uniform Continuity

Chapter 5 Differentiation of Functions of One Variable

§ 1. Derivative of a Function and Its Computation

76. Problem of Calculating the Velocity of a Moving Point

77. Problem of Constructing a Tangent to a Curve

78. Definition of the Derivative

79. Examples of the Calculation of the Derivative

80. Derivative of the Inverse Function

81. Summary of Formula for Derivatives

82. Formula for the Increment of a Function

83. Rules for the Calculation of Derivatives

84. Derivative of a Compound Function

85. Examples

86. One-Sided Derivatives

87. Infinite Derivatives

88. Further Examples of Exceptional Cases

§ 2. The Differential

89. Definition of the Differential

90. The Relation between the Differentiability and the Existence of the Derivative

91. Fundamental Formula and Rules of Differentiation

92. Invariance of the Form of the Differential

93. Differentials as a Source of Approximate Formula

94. Application of Differentials in Estimating Errors

§ 3. Derivatives and Differentials of Higher Orders

95. Definition of Derivatives of Higher Orders

96. General Formula for Derivatives of Arbitrary Order

97. The Leibniz Formula

98. Differentials of Higher Orders

99. Violation of the Invariance of the Form for Differentials of Higher Orders

Chapter 6 Basic Theorems of Differential Calculus

§ 1. Mean Value Theorems

100. Fermat's Theorem

101. Rolle's Theorem

102. Theorem on Finite Increments

103. The Limit of the Derivative

104. Generalized Theorem on Finite Increments

§ 2. Taylor's Formula

105. Taylor's Formula for a Polynomial

106. Expansion of an Arbitrary Function

107. Another Form for the Remainder Term

108. Application of the Derived Formula to Elementary Functions

109. Approximate Formula. Examples

Chapter 7 Investigation of Functions by Means of Derivatives

§ 1. Investigation of the Behavior of Functions

110. Conditions That a Function May Be Constant

111. Condition of Monotonicity of a Function

112. Maxima and Minima; Necessary Conditions

113. The First Rule

114. The Second Rule

115. Construction of the Graph of a Function

116. Examples

117. Application of Higher Derivatives

§ 2. The Greatest and the Smallest Values of a Function

118. Determination of the Greatest and the Smallest Values

119. Problems

§ 3. Solution of Indeterminate Forms

120. Indeterminate Forms of the Type 0/0

121. Indeterminate Forms of the Type ∞/∞

122. Other Types of Indeterminate Forms

Chapter 8 Functions of Several Variables

§ 1. Basic Concepts

123. Functional Dependence between Variables. Examples

124. Functions of Two Variables and Their Domains of Definition

125. Arithmetic M-Dimensional Space

126. Examples of Domains in M-Dimensional Space

127. General Definition of Open and Closed Domains

128. Function of M Variables

129. Limit of a Function of Several Variables

130. Examples

131. Repeated Limits

§ 2. Continuous Functions

132. Continuity and Discontinuities of Functions of Several Variables

133. Operations on Continuous Functions

134. Theorem on the Vanishing of a Function

135. The Bolzano-Weierstrass Lemma

136. Theorem on the Boundedness of a Function

137. Uniform Continuity

Chapter 9 Differentiation of Functions of Several Variables

§ 1. Derivatives and Differentials of Functions of Several Variables

138. Partial Derivatives

139. Total Increment of the Function

140. Derivatives of Compound Functions

141. Examples

142. The Total Differential

143. Invariance of the Form of the (First) Differential

144. Application of the Total Differential to Approximate Calculations

145. Homogeneous Functions

§ 2. Derivatives and Differentials of Higher Orders

146. Derivatives of Higher Orders

147. Theorems on Mixed Derivatives

148. Differentials of Higher Orders

149. Differentials of Compound Functions

150. The Taylor Formula

§ 3 . Extrema, the Greatest and the Smallest Values

151. Extrema of Functions of Several Variables. Necessary Conditions

152. Investigation of Stationary Points (For the Case of Two Variables)

153. The Smallest and the Greatest Values of a Function. Examples

154. Problems

Chapter 10 Primitive Functions (Indefinite Integral)

§ 1. Indefinite Integral and Simple Methods for Its Evaluation

155. The Concept of a Primitive Function (and of an Indefinite Integral)

156. The Integral and the Problem of Determination of Area

157. Collection of the Basic Integrals

158. Rules of Integration

159. Examples

160. Integration by a Change of Variable

161. Examples

162. Integration by Parts

163. Examples

§ 2 . Integration of Rational Expressions

164. Formulation of the Problem of Integration in Finite Form

165. Simple Fractions and Their Integration

166. Integration of Proper Fractions

167. Ostrogradski's Method for Separating the Rational Part of an Integra

§ 3. Integration of Some Expressions Containing Roots

168. Integration of Expression of the Form R

169. Integration of Binomial Differentials

170. Integration of Expressions of the Form R[X, √(Ax2 + Bx + C)] Euler's Substitution

§ 4. Integration of Expressions Containing Trigonometric and Exponential Functions

171. Integration of the Differentials R(Sin X, Cos X)Dx

172. Survey of Other Cases

§ 5. Elliptic Integrals

173. Definitions

174. Reduction to the Canonical Form

Chapter 11 Definite Integral

§ 1. Definition and Conditions for the Existence of a Definite Integral

175. Another Formulation of the Area Problem

176. Definition

177. Darboux's Sums

178. Condition for the Existence of the Integral

179. Classes of Integrable Functions

§ 2. Properties of Definite Integrals

180. Integrals over an Oriented Interval

181. Properties Expressed by Equalities

182. Properties Expressed by Inequalities

183. Definite Integral as a Function of the Upper Limit

§ 3. Evaluation and Transformation of Definite Integrals

184. Evaluation Using Integral Sums

185. The Fundamental Formula of Integral Calculus

186. The Formula for the Change of Variable in a Definite Integral

187. Integration by Parts in a Definite Integral

188. Wallis's Formula

§ 4. Approximate Evaluation of Integrals

189. The Trapezium Formula

190. Parabolic Formula

191. Remainder Term for the Approximate Formula

192. Example

Chapter 1 2 Geometric and Mechanical Applications of the Integral Calculus

§ 1. Areas and Volumes

193. Definition of the Concept of Area. Quadrable Domains

194. The Additive Property of Area

195. Area as a Limit

196. An Integral Expression for Area

197. Definition of the Concept of Volume and Its Properties

198. Integral Expression for the Volume

§ 2. Length of Arc

199. Definition of the Concept of the Length of an Arc

200. Lemmas

201. Integral Expression for the Length of an Arc

202. Variable Arc and Its Differential

203. Length of the Arc of a Spatial Curve

§ 3. Computation of Mechanical and Physical Quantities

204. Applications of Definite Integrals

205. The Area of a Surface of Revolution

206. Calculation of Static Moments and Center of Mass of a Curve

207. Determination of Static Moments and Center of Mass of a Plane Figure

208. Mechanical Work

Chapter 13 Some Geometric Applications of the Differential Calculus

§ 1. The Tangent and the Tangent Plane

209. Analytic Representation of Plane Curves

210. Tangent to a Plane Curve

211. Positive Direction of the Tangent

212. The Case of a Spatial Curve

213. The Tangent Plane to a Surface

§ 2. Curvature of a Plane Curve

214. The Direction of Concavity, Points of Inflection

215. The Concept of Curvature

216. The Circle of Curvature and Radius of Curvature

Chapter 14 Historical Survey of the Development of the Fundamental Concepts of Mathematical Analysis

§ 1. Early History of the Differential and Integral Calculus

217. Seventeenth Century and the Analysis of Infinitesimals

218. The Method of Indivisibles

219. Further Development of the Science of Indivisibles

220. Determination of the Greatest and Smallest Quantities; Construction of Tangents

221. Construction of Tangents by Means of Kinematic Considerations

222. Mutual Invertibility of the Problems of Construction of Tangent and Squaring

223. Survey of the Foregoing Achievements

§ 2. Isaac Newton (1642-1727)

224. The Calculus of Fluxions

225. The Calculus Inverse to the Calculus of Fluxions; Squaring

226. Newton's Principles and the Origin of the Theory of Limits

227. Problems of Foundations in Newton's Works

§ 3. Gottfried Wilhelm Leibniz (1646-1716)

228. First Steps in Creating the New Calculus

229. The First Published Work on Differential Calculus

230. The First Published Paper on Integral Calculus

231. Further Works of Leibniz. Creation of a School

232. Problems of Foundation in Leibniz's Works

233. Postscript

Index

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The Fundamentals of Mathematical Analysis

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