The Fundamentals of Mathematical Analysis

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