Fikhtengol'ts / Sneddon / Stark The Fundamentals of Mathematical Analysis

1. Auflage 2014
ISBN: 978-1-4831-5413-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 540 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

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 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.

Autoren/Hrsg.

Fikhtengol'ts, G. M.

Sneddon, I. N.

Stark, M.

Ulam, S.

Inhaltsverzeichnis

1;Front Cover;1
2;The Fundamentals of Mathematical Analysis;4
3;Copyright Page;5
4;Table of Contents;6
5;CHAPTER 15. SERIES OF NUMBERS;24
5.1;§ 1. Introduction;24
5.2;§ 2. The convergence of positive series;29
5.3;§ 3. The convergence of arbitrary series;44
5.4;§ 4. The properties of convergent series;50
5.5;§ 5. Infinite products;60
5.6;§ 6. The expansion of elementary functions in power series;67
5.7;§ 7. Approximate calculations using series;81
6;CHAPTER 16. SEQUENCES AND SERIES OF FUNCTIONS;88
6.1;§ 1. Uniform convergence;88
6.2;§ 2. The functional properties of the sum of a series;96
6.3;§ 3. Power series and series of polynomials;109
6.4;§ 4. An outline of the history of series;123
7;CHAPTER 17. IMPROPER INTEGRALS;133
7.1;§ 1. Improper integrals with infinite limits;133
7.2;§ 2. Improper integrals of unbounded functions;144
7.3;§ 3. Transformation and evaluation of improper integrals;151
8;CHAPTER 18. INTEGRALS DEPENDING ON A PARAMETER;159
8.1;§ 1. Elementary theory;159
8.2;§ 2. Uniform convergence of integrals;170
8.3;§ 3. The use of the uniform convergence of integrals;177
8.4;§ 4. Eulerian integrals;191
9;CHAPTER 19. IMPLICIT FUNCTIONS. FUNCTIONAL DETERMINANTS;204
9.1;§ 1. Implicit functions;204
9.2;§ 2. Some applications of the theory of implicit functions;222
9.3;§ 3. Functional determinants and their formal properties;236
10;CHAPTER 20. CURVILINEAR INTEGRALS;242
10.1;§ 1. Curvilinear integrals of the first kind;242
10.2;§ 2. Curvilinear integrals of the second kind;249
11;CHAPTER 21. DOUBLE INTEGRALS;266
11.1;§ 1. The definition and simplest properties of double integrals;266
11.2;§ 2. The evaluation of a double integral;282
11.3;§ 3 . Greenes formula;298
11.4;§ 4. Conditions for a curvilinear integral to be independent of the path of integration;304
11.5;§ 5. Change of variables in double integrals;315
12;CHAPTER 22. THE AREA OF A SURFACE. SURFACE INTEGRALS;338
12.1;§ 1. Two-sided surfaces;338
12.2;§ 2. The area of a curved surface;351
12.3;§ 3. Surface integrals of the first type;361
12.4;§ 4. Surface integrals of the second type;367
13;CHAPTER 23. TRIPLE INTEGRALS;380
13.1;§ 1. A triple integral and its evaluation;380
13.2;§ 2. Ostrogradski's formula;391
13.3;§ 3. Change of variables in triple integrals;398
13.4;§ 4. The elementary theory of a field;411
13.5;§ 5. Multiple integrals;423
14;CHAPTER 24. FOURIER SERIES;427
14.1;§ 1. Introduction;427
14.2;§ 2. The expansion of functions in Fourier series;435
14.3;§ 3. The Fourier integral;458
14.4;§ 4. The closed and complete nature of a trigonometrical system of functions;468
14.5;§ 5. An outline of the history of trigonometrical series;485
15;CONCLUSION AN OUTLINE OF FURTHER DEVELOPMENTS IN MATHEMATICAL ANALYSIS;499
15.1;I. The theory of differential equations;499
15.2;II. Variational calculus;502
15.3;III. The theory of functions of a complex variable;507
15.4;IV. The theory of integral equations;511
15.5;V. The theory of functions of a real variable;515
15.6;VI. Functional analysis;521
16;Index;530
17;Other Titles in the Series;540