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Mathematical Analysis

Differentiation and Integration

1st Edition - January 1, 1965

Authors: I.G. Aramanovich, R. S. Guter, L.A. Lyusternik

Editor: I. N. Sneddon

eBook ISBN:

9 7 8 - 1 - 4 8 3 1 - 5 6 2 4 - 8

Mathematical Analysis: Differentiation and Integration is devoted to two basic operations of mathematical analysis, differentiation and integration. The problems directly… Read more

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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.

Foreword

Notation

I The Differentiation of Functions of One Variable

§ 1 Derivatives and Differentials of the First Order

§ 2 Derivatives and Differentials of Higher Orders. Taylor's Series

§ 3 The Application of Derivatives in Investigating Functions. Extrema

§ 4 Differential Operators

II The Differentiation of Functions of n Variables

§ 1 Derivatives and Differentials of the First Order

§ 2 Derivatives and Differentials of Higher Orders. Taylor's Series

§ 3 Polynomials of Differential Operators

§ 4 The Differentiation of Mappings from En into Em

§ 5 Extrema

§ 6 Stationary Points

III Composite and Implicit Functions of n Variables

§ 1 Transformation of Variables. Composite Functions

§ 2 Implicit Functions. Functions Depending on a Parameter

§ 3 Newton's Diagram

§ 4 The Representation of Functions of n Variables in the Form of Superpositions

IV Systems of Functions and Curvilinear Coordinates in a Plane and in Space

§ 1 Mapping. Jacobians

§ 2 Curvilinear Coordinates in a Plane

§ 3 Curvilinear Coordinates in Space

V The Integration of Functions

§ 1 The Indefinite Integral

§ 2 The Integration of Elementary Functions

§ 3 The Definite Integral

§ 4 The Integration of Functions of n Variables

§ 5 The Application of Definite Integrals to Problems of Geometry and Mechanics

VI Improper Integrals. Integrals Dependent on a Parameter. Stieltjes’ Integral

§ 1 Improper Integrals

§ 2 Limiting Process under the Sign of the Integral. Integrals Dependent on a Parameter

§ 3 The Stieltjes Integral for Functions of One Variable

§ 4 Integrals and Derivatives of Fractional Orders

VII The Transformation of Differential and Integral Expressions

§ 1 Transformation of Differential Expressions

§ 2 Transformation of Integral Expressions

§ 3 Formulae for Transformation of Integrals

Appendixes

1. Derivatives of Elementary Functions

2. The Expansion of Elementary Functions into Power Series

3. Integrals of Elementary Functions

4. Special Functions Defined by Integrals

References

Index

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