The Fundamentals of Mathematical Analysis

The Fundamentals of Mathematical Analysis, Volume 1 is a textbook that provides a systematic and rigorous treatment of the fundamentals of mathematical analysis. Emphasis is placed on the concept of limit which plays a principal role in mathematical analysis. Examples of the application of mathematical analysis to geometry, mechanics, physics, and engineering are given. This volume is comprised of 14 chapters and begins with a discussion on real numbers, their properties and applications, and arithmetical operations over real numbers. The reader is then introduced to the concept of function, important classes of functions, and functions of one variable; the theory of limits and the limit of a function, monotonic functions, and the principle of convergence; and continuous functions of one variable. A systematic account of the differential and integral calculus is then presented, paying particular attention to differentiation of functions of one variable; investigation of the behavior of functions by means of derivatives; functions of several variables; and differentiation of functions of several variables. The remaining chapters focus on the concept of a primitive function (and of an indefinite integral); definite integral; geometric applications of integral and differential calculus. This book is intended for first- and second-year mathematics students.

Volume I

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 an 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 Function (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 Integral

§ 3. Integration of Some Expressions Containing Roots

168. Integration of Expressions of the Form R[x,m√(αx+ß/γx+δ)]

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

111. 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 12 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

Other Titles in the Series

Volume II

Chapter 15 Series of Numbers

§ 1. Introduction

234. Elementary Concepts

235. The Most Elementary Theorems

§ 2. The Convergence of Positive Series

236. A Condition for the Convergence of a Positive Series

237. Theorems on the Comparison of Series

238. Examples

239. Cauchy's and d'Alembert's Tests

240. Raabe's Test

241. The Maclaurin-Cauchy Integral Test

§ 3. The Convergence of Arbitrary Series

242. The Principle of Convergence

243. Absolute Convergence

244. Alternating Series

§ 4. The Properties of Convergent Series

245. The Associative Property

246. The Permuting Property of Absolutely Convergent Series

247. The Case of Non-Absolutely Convergent Series

248. The Multiplication of Series

§ 5. Infinite Products

249. Fundamental Concepts

250. The Simplest Theorems. The Connection with Series

251. Examples

§ 6. The Expansion of Elementary Functions in Power Series

252. Taylor Series

253. The Expansion of the Exponential and Elementary Trigonometrical Functions in Power Series

254. Euler's Formula

255. The Expansion for the Inverse Tangent

256. Logarithmic Series

257. Stirling's Formula

258. Binomial Series

259. A Remark on the Study of the Remainder

§ 7. Approximate Calculations Using Series

260. Statement of the Problem

261. The Calculation of the Number π

262. The Calculation of Logarithms

Chapter 16 Sequences and Series of Functions

§ 1. Uniform Convergence

263. Introductory Remarks

264. Uniform and Non-Uniform Convergence

265. The Condition for Uniform Convergence

§ 2. The Functional Properties of the Sum of a Series

266. The Continuity of the Sum of a Series

267. The Case of Positive Series

268. Termwise Transition to a Limit

269. Termwise Integration of Series

270. Termwise Differentiation of Series

271. An Example of a Continuous Function without a Derivative

§ 3. Power Series and Series of Polynomials

272. The Interval of Convergence of a Power Series

273. The Continuity of the Sum of a Power Series

274. Continuity at the End Points of the Interval of Convergence

275. Termwise Integration of a Power Series

276. Termwise Differentiation of a Power Series

277. Power Series as Taylor Series

278. The Expansion of a Continuous Function in a Series of Polynomials

§ 4. An Outline of the History of Series

279. The Epoch of Newton and Leibniz

280. The Period of the Formal Development of the Theory of Series

281. The Creation of a Precise Theory

Chapter 17 Improper Integrals

§ 1. Improper Integrals with Infinite Limits

282. The Definition of Integrals with Infinite Limits

283. The Application of the Fundamental Formula of Integral Calculus

284. An Analogy with Series. Some Simple Theorems

285. The Convergence of the Integral in the Case of a Positive Function

286. The Convergence of the Integral in the General Case

287. More Refined Tests

§ 2. Improper Integrals of Unbounded Functions

288. The Definition of Integrals of Unbounded Functions

289. An Application of the Fundamental Formula of Integral Calculus

290. Conditions and Tests for the Convergence of an Integral

§ 3. Transformation and Evaluation of Improper Integrals

291. Integration by Parts in the Case of Improper Integrals

292. Change of Variables in Improper Integrals

293. The Evaluation of Integrals by Artificial Methods

Chapter 18 Integrals Depending on a Parameter

§ 1. Elementary Theory

294. Statement of the Problem

295. Uniform Approach to a Limit Function

296. Taking Limits under the Integral Sign

297. Differentiation under the Integral Sign

298. Integration under the Integral Sign

299. The Case When the Limits of the Integral also Depend on the Parameter

300. Examples

§ 2. Uniform Convergence of Integrals

301. The Definition of Uniform Convergence of Integrals

302. Conditions and Sufficiency Tests for Uniform Convergence

303. The Case of Integrals with Finite Limits

§ 3. The Use of the Uniform Convergence of Integrals

304. Taking Limits under the Integral Sign

305. The Integration of an Integral with Respect to the Parameter

306. Differentiation of an Integral with Respect to the Parameter

307. A Remark on Integrals with Finite Limits

308. The Evaluation of Some Improper Integrals

§ 4. Eulerian Integrals

309. The Eulerian Integral of the First Type

310. The Eulerian Integral of the Second Type

311. Some Simple Properties of the Γ Function

312. Examples

313. Some Historical Remarks on Changing the Order of Two Limit Operations

Chapter 19 Implicit Functions. Functional Determinants

§ 1. Implicit Functions

314. The Concept of an Implicit Function of One Variable

315. The Existence and Properties of an Implicit Function

316. An Implicit Function of Several Variables

317. The Determination of Implicit Functions From a System of Equations

318. The Evaluation of Derivatives of Implicit Functions

§ 2. Some Applications of the Theory of Implicit Functions

319. Relative Extremes

320. Lagrange's Method of Undetermined Multipliers

321. Examples and Problems

322. The Concept of the Independence of Functions

323. The Rank of a Functional Matrix

§ 3. Functional Determinants and Their Formal Properties

324. Functional Determinants

325. The Multiplication of Functional Determinants

326. The Multiplication of Non-Square Functional Matrices

Chapter 20 Curvilinear Integrals

§ 1. Curvilinear Integrals of the First Kind

327. The Definition of a Curvilinear Integral of the First Kind

328. The Reduction to an Ordinary Definite Integral

329. Examples

§ 2. Curvilinear Integrals of the Second Kind

330. The Definition of Curvilinear Integrals of the Second Kind

331. The Existence and Evaluation of a Curvilinear Integral of the Second Kind

332. The Case of a Closed Contour. The Orientation of the Plane

333. Examples

334. The Connection between Curvilinear Integrals of Both Kinds

335. Applications to Physical Problems

Chapter 21 Double Integrals

§ 1. The Definition and Simplest Properties of Double Integrals

336. The Problem of the Volume of a Cylindrical Body

337. The Reduction of a Double Integral to a Repeated Integral

338. The Definition of a Double Integral

339. A Condition for the Existence of a Double Integral

340. Classes of Integrable Functions

341. The Properties of Integrable Functions and Double Integrals

342. An Integral as an Additive Function of the Domain; Differentiation in the Domain

§ 2. The Evaluation of a Double Integral

343. The Reduction of a Double Integral to a Repeated Integral in the Case of a Rectangular Domain

344. The Reduction of a Double Integral to a Repeated Integral in the Case of a Curvilinear Domain

345. A Mechanical Application

§ 3. Green's Formula

346. The Derivation of Green's Formula

347. An Expression for Area By Means of Curvilinear Integrals

§ 4. Conditions for a Curvilinear Integral to Be Independent of the Path of Integration

348. The Integral along a Simple Closed Contour

349. The Integral along a Curve Joining Two Arbitrary Points

350. The Connection with the Problem of Exact Differentials

351. Applications to Physical Problems

§ 5. Change of Variables in Double Integrals

352. Transformation of Plane Domains

353. An Expression for Area in Curvilinear Coordinates

354. Additional Remarks

355. A Geometrical Derivation

356. Change of Variables in Double Integrals

357. The Analogy with a Simple Integral. The Integral over an Oriented Domain

358. Examples

359. Historical Note

Chapter 22 The Area of a Surface. Surface Integrals

§ 1. Two-Sided Surfaces

360. Parametric Representation of a Surface

361. The Side of a Surface

362. The Orientation of a Surface and the Choice of a Side of It

363. The Case of a Piece-Wise Smooth Surface

§ 2. The Area of a Curved Surface

364. Schwarz's Example

365. The Area of a Surface Given By an Explicit Equation

366. The Area of a Surface in the General Case

367. Examples

§ 3. Surface Integrals of the First Type

368. The Definition of a Surface Integral of the First Type

369. The Reduction to an Ordinary Double Integral

370. Mechanical Applications of Surface Integrals of the First Type

§ 4. Surface Integrals of the Second Type

371. The Definition of Surface Integrals of the Second Type

372. The Reduction to an Ordinary Double Integral

373. Stokes's Formula

374. The Application of Stokes's Formula to the Investigation of Curvilinear Integrals in Space

Chapter 23 Triple Integrals

§ 1. A Triple Integral and Its Evaluation

375. The Problem of Calculating the Mass of a Solid

376. A Triple Integral and the Conditions for Its Existence

377. The Properties of Integrable Functions and Triple Integrals

378. The Evaluation of a Triple Integral

379. Mechanical Applications

§ 2. Ostrogradski's Formula

380. Ostrogradski's Formula

381. Some Examples of Applications of Ostrogradski's Formula

§ 3. Change of Variables in Triple Integrals

382. The Transformation of Space Domains

383. An Expression for Volume in Curvilinear Coordinates

384. A Geometrical Derivation

385. Change of Variables in Triple Integrals

386. Examples

387. Historical Note

§ 4. The Elementary Theory of a Field

388. Scalars and Vectors

389. Scalar and Vector Fields

390. A Derivative in a Given Direction. Gradient

391. The Flow of a Vector Through a Surface

392. Ostrogradski's Formula. Divergence

393. The Circulation of a Vector. Stokes's Formula. Vortex

§ 5. Multiple Integrals

394. The Volume of an m-Dimensional Body and the m-Tuple Integral

395. Examples

Chapter 24 Fourier Series

§ 1. Introduction

396. Periodic Values and Harmonic Analysis

397. The Determination of Coefficients by the Euler-Fourier Method

398. Orthogonal Systems of Functions

§ 2. The Expansion of Functions in Fourier Series

399. Statement of the Problem. Dirichlet's Integral

400. A Fundamental Lemma

401. The Principle of Localization

402. The Representation of a Function By Fourier Series

403. The Case of a Non-Periodic Function

404. The Case of an Arbitrary Interval

405. An Expansion in Cosines Only, or in Sines Only

406. Examples

407. The Expansion of a Continuous Function in a Series of Trigonometrical Polynomials

§ 3. The Fourier Integral

408. The Fourier Integral as a Limiting Case of a Fourier Series

409. Preliminary Remarks

410. The Representation of a Function by a Fourier Integral

411. Different Forms of Fourier's Formula

412. Fourier Transforms

§ 4. The Closed and Complete Nature of a Trigonometrical System of Functions

413. Mean Approximation to Functions

414. The Closure of a Trigonometrical System

415. The Completeness of a Trigonometrical System

416. The Generalized Equation of Closure

417. Termwise Integration of a Fourier Series

418. The Geometrical Interpretation

§ 5. An Outline of the History of Trigonometrical Series

419. The Problem of the Vibration of a String

420. D'Alembert's and Euler's Solution

421. Taylor's and D. Bernoulli's Solution

422. The Controversy Concerning the Problem of the Vibration of a String

423. The Expansion of Functions in Trigonometrical Series; the Determination of Coefficients

424. The Proof of the Convergence of Fourier Series and Other Problems

425. Concluding Remarks

Conclusion An Outline of Further Developments in Mathematical Analysis

I. The Theory of Differential Equations

II. Variational Calculus

III. The Theory of Functions of a Complex Variable

IV. The Theory of Integral Equations

V. The Theory of Functions of a Real Variable

VI. Functional Analysis

Index

Other Titles in the Series

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The Fundamentals of Mathematical Analysis

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