The Fundamentals of Mathematical Analysis
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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.
Table of Contents
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
Product details
- No. of pages: 520
- Language: English
- Copyright: © Pergamon 1965
- Published: January 1, 1965
- Imprint: Pergamon
- eBook ISBN: 9781483139074
About the Editor
I. N. Sneddon
About the Author
G. M. Fikhtengol'ts
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