Aramanovich / Guter / Lyusternik Mathematical Analysis

Differentiation and Integration
1. Auflage 2014
ISBN: 978-1-4831-5624-8
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 334 Seiten, Web PDF

Mathematical Analysis: Differentiation and Integration is devoted to two basic operations of mathematical analysis, differentiation and integration. The problems directly connected with the operations of differentiation and integration of functions of one or several variables are discussed, together with elementary generalizations of these operations. This volume is comprised of seven chapters and begins by considering the differentiation of functions of one variable and of n variables, paying particular attention to derivatives and differentials as well as their properties. The next chapter deals with composite and implicit functions of n variables in connection with differentiation, along with the representation of functions in the form of superpositions. Subsequent chapters offer detailed accounts of systems of functions and curvilinear coordinates in a plane and in space; the integration of functions; and improper integrals. The final chapter examines the transformation of differential and integral expressions. This book will be a useful resource for mathematicians and mathematics students.

Autoren/Hrsg.

Aramanovich, I. G.

Guter, R. S.

Lyusternik, L. A.

Sneddon, I. N.

Inhaltsverzeichnis

1;Front Cover;1
2;Mathematical Analysis: Differentiation and Integration;4
3;Copyright Page;5
4;Table of Contents;6
5;FOREWORD;8
6;NOTATION;10
7;CHAPTER 1. THE DIFFERENTIATION OF FUNCTIONS OF ONE VARIABLE;14
7.1;§ 1 Derivatives and Differentials of the First Order;14
7.2;§ 2 Derivatives and Differentials of Higher Orders. Taylor's Series;34
7.3;§ 3 The Application of Derivatives in Investigating Functions. Extrema;45
7.4;§ 4 Differential Operators;53
8;CHAPTER 2. THE DIFFERENTIATION OF FUNCTIONS OF n VARIABLES;59
8.1;§ 1 Derivatives and Differentials of the First Order;59
8.2;§ 2 Derivatives and Differentials of Higher Orders. Taylor's Series;68
8.3;§ 3 Polynomials of Differential Operators;74
8.4;§ 4 The Differentiation of Mappings from En into Em;76
8.5;§ 5 Extrema;78
8.6;§ 6 Stationary Points;85
9;CHAPTER 3. COMPOSITE AND IMPLICIT FUNCTIONS OF n VARIABLES;89
9.1;§ 1 Transformation of Variables. Composite Functions;89
9.2;§ 2 Implicit Functions. Functions Depending on a Parameter;94
9.3;§ 3 Newton's Diagram;102
9.4;§ 4 The Representation of Functions of n Variables in the Form of Superpositions;108
10;CHAPTER 4. SYSTEMS OF FUNCTIONS AND CURVILINEAR COORDINATES IN A PLANE AND IN SPACE;112
10.1;§ 1 Mapping. Jacobians;112
10.2;§ 2 Curvilinear Coordinates in a Plane;121
10.3;§ 3 Curvilinear Coordinates in Space;132
11;CHAPTER 5. THE INTEGRATION OF FUNCTIONS;148
11.1;§ 1 The Indefinite Integral;148
11.2;§2 The Integration of Elementary Functions;150
11.3;§ 3 The Definite Integral;166
11.4;§ 4 The Integration of Functions of n Variables;176
11.5;§ 5 The Application of Definite Integrals to Problems of Geometry and Mechanics;189
12;CHAPTER 6. IMPROPER INTEGRALS. INTEGRALS DEPENDENT ON A PARAMETER. STIELTJES' INTEGRAL;205
12.1;§ 1 Improper Integrals;205
12.2;§ 2 Limiting Process under the Sign of the Integral. Integrals Dependent on a Parameter;225
12.3;§ 3 The Stieltjes Integral for Functions of One Variable;233
12.4;§ 4 Integrals and Derivatives of Fractional Orders;237
13;CHAPTER 7. THE TRANSFORMATION OF DIFFERENTIAL AND INTEGRAL EXPRESSIONS;239
13.1;§ 1 Transformation of Differential Expressions;239
13.2;§ 2 Transformation of Integral Expressions;254
13.3;§ 3 Formulae for Transformation of Integrals;262
14;APPENDIXES;270
14.1;1. Derivatives of Elementary Functions;270
14.2;2. The Expansion of Elementary Functions into Power Series;273
14.3;3. Integrals of Elementary Functions;275
14.4;4. Special Functions Defined by Integrals;297
15;REFERENCES;322
16;INDEX;324
17;OTHER TITLES IN THE SERIES;333