Fikhtengol'ts / Sneddon The Fundamentals of Mathematical Analysis

1. Auflage 2014
ISBN: 978-1-4831-3907-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 72, 520 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

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.

Autoren/Hrsg.

Fikhtengol'ts, G. M.

Sneddon, I. N.

Inhaltsverzeichnis

1;Front Cover;1
2;The Fundamentals of Mathematical Analysis;4
3;Copyright Page;5
4;Table of Contents;6
5;Foreword;24
6;CHAPTER 1. REAL NUMBERS;28
6.1;§ 1. The set of real numbers and its ordering;28
6.2;§ 2. Arithmetical operations over real numbers;41
6.3;§ 3. Further properties and applications of real numbers;45
7;CHAPTER 2. FUNCTIONS OF ONE VARIABLE;52
7.1;§ 1. The concept of a function;52
7.2;§ 2. Important classes of functions;66
8;CHAPTER 3. THEORY OF LIMITS;79
8.1;§1. The limit of a function;79
8.2;§ 2. Theorems on limits;99
8.3;§ 3. Monotonic functions;115
8.4;§ 4. The number e;122
8.5;§ 5. The principle of convergence;129
8.6;§ 6. Classification of infinitely small and infinitely large quantities;135
9;CHAPTER 4. CONTINUOUS FUNCTIONS OF ONE VARIABLE;142
9.1;§ 1. Continuity (and discontinuity) of a function;142
9.2;§ 2. Properties of continuous functions;154
10;CHAPTER 5. DIFFERENTIATION OF FUNCTIONS OF ONE VARIABLE;167
10.1;§ 1. Derivative of a function and its computation;167
10.2;§ 2. The differential;192
10.3;§ 3. Derivatives and differentials of higher orders;200
11;CHAPTER 6. BASIC THEOREMS OF DIFFERENTIAL CALCULUS;210
11.1;§ 1. Mean value theorems;210
11.2;§ 2. Taylor's formula;218
12;CHAPTER 7. INVESTIGATION OF FUNCTIONS BY MEANS OF DERIVATIVES;231
12.1;§ 1. Investigation of the behaviour of functions;231
12.2;§ 2. The greatest and the smallest values of a function;245
12.3;§ 3. Solution of indeterminate forms;248
13;CHAPTER 8. FUNCTIONS OF SEVERAL VARIABLES;256
13.1;§ 1. Basic concepts;256
13.2;§ 2. Continuous functions;276
14;CHAPTER 9. DIFFERENTIATION OF FUNCTIONS OF SEVERAL VARIABLES;285
14.1;§ 1. Derivatives and differentials of functions of several variables;285
14.2;§ 2. Derivatives and differentials of higher orders;302
14.3;§ 3. Extrema, the greatest and the smallest values;313
15;CHAPTER 10. PRIMITIVE FUNCTION (INDEFINITE INTEGRAL);326
15.1;§ 1. Indefinite integral and simple methods for its evaluation;326
15.2;§ 2. Integration of rational expressions;345
15.3;§ 3. Integration of some expressions containing roots;354
15.4;§ 4. Integration of expressions containing trigonometric and exponential functions;363
15.5;§ 5. Elliptic integrals;368
16;CHAPTER 11. DEFINITE INTEGRAL;371
16.1;§ 1. Definition and conditions for the existence of a definite integral;371
16.2;§ 2. Properties of definite integrals;381
16.3;§ 2. Properties of definite integrals;381
16.4;§ 4. Approximate evaluation of integrals;398
17;CHAPTER 12. GEOMETRIC AND MECHANICAL APPLICATIONS OF THE INTEGRAL CALCULUS;408
17.1;§ 1. Areas and volumes;408
17.2;§ 2. Length of arc;426
17.3;§ 3. Computation of mechanical and physical quantities;436
18;CHAPTER 13. SOME GEOMETRIC APPLICATIONS OF THE DIFFERENTIAL CALCULUS;450
18.1;§ 1. The tangent and the tangent plane;450
18.2;§ 2. Curvature of a plane curve;465
19;CHAPTER 14. HISTORICAL SURVEY OF THE DEVELOPMENT OF THE FUNDAMENTAL CONCEPTS OF MATHEMATICAL ANALYSIS;475
19.1;§ 1. Early history of the differential and integral calculus;475
19.2;§ 2. Isaac Newton (1642-1727);490
19.3;§ 3. Gottfried Wilhelm Leibniz (1646-1716);500
20;Index;510
21;Other Titles in the Series;520