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The Fundamentals of Mathematical Analysis
- 1st Edition - January 1, 1965
- Author: G. M. Fikhtengol'ts
- Editors: I. N. Sneddon, M. Stark, S. Ulam
- Language: English
- Paperback ISBN:9 7 8 - 1 - 4 8 3 1 - 2 1 7 1 - 0
- Hardback ISBN:9 7 8 - 0 - 0 8 - 0 1 0 0 6 0 - 9
- eBook ISBN:9 7 8 - 1 - 4 8 3 1 - 5 4 1 3 - 8
The Fundamentals of Mathematical Analysis, Volume 2 is a continuation of the discussion of the fundamentals of mathematical analysis, specifically on the subject of curvilinear and… Read more
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Request a sales quoteThe Fundamentals of Mathematical Analysis, Volume 2 is a continuation of the discussion of the fundamentals of mathematical analysis, specifically on the subject of curvilinear and surface integrals, with emphasis on the difference between the curvilinear and surface ""integrals of first kind"" and ""integrals of second kind."" The discussions in the book start with an introduction to the elementary concepts of series of numbers, infinite sequences and their limits, and the continuity of the sum of a series. The definition of improper integrals of unbounded functions and that of uniform convergence of integrals are explained. Curvilinear integrals of the first and second kinds are analyzed mathematically. The book then notes the application of surface integrals, through a parametric representation of a surface, and the calculation of the mass of a solid. The text also highlights that Green's formula, which connects a double integral over a plane domain with curvilinear integral along the contour of the domain, has an analogue in Ostrogradski's formula. The periodic values and harmonic analysis such as that found in the operation of a steam engine are analyzed. The volume ends with a note of further developments in mathematical analysis, which is a chronological presentation of important milestones in the history of analysis. The book is an ideal reference for mathematicians, students, and professors of calculus and advanced mathematics.
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. The 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 W-Dimensional Body and the W-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 Series 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. Extreme Properties of a Fourier Series!
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
- No. of pages: 540
- Language: English
- Edition: 1
- Published: January 1, 1965
- Imprint: Pergamon
- Paperback ISBN: 9781483121710
- Hardback ISBN: 9780080100609
- eBook ISBN: 9781483154138