Integral and measure : from rather simple to rather complex / Vigirdas Mackevicius.

Format:

Book

Author/Creator:

Mackevičius, Vigirdas, author.

Series:

Mathematics and statistics series (ISTE)

Mathematics and Statistics Series

Language:

English

Subjects (All):

Measure theory.

Physical Description:

1 online resource (302 p.)

Edition:

1st ed.

Place of Publication:

London, England ; Hoboken, New Jersey : ISTE : Wiley, 2014.

Language Note:

English

Summary:

This book is devoted to integration, one of the two main operations in calculus. In Part 1, the definition of the integral of a one-variable function is different (not essentially, but rather methodically) from traditional definitions of Riemann or Lebesgue integrals. Such an approach allows us, on the one hand, to quickly develop the practical skills of integration as well as, on the other hand, in Part 2, to pass naturally to the more general Lebesgue integral. Based on the latter, in Part 2, the author develops a theory of integration for functions of several variables. In Part 3, within

Contents:

Cover page; Half-title page; Title page; Copyright page; Contents; Preface; Note for the Teacher or Who is better, Riemann or Lebesgue?; Notation; Part 1: Integration of One-Variable Functions; 1: Functions without Second-kind Discontinuities; P.1. Problems; 2: Indefinite Integral; P.2. Problems; 3: Definite Integral; 3.1. Introduction; P.3. Problems; 4: Applications of the Integral; 4.1. Area of a curvilinear trapezium; 4.2. A general scheme for applying the integrals; 4.3. Area of a surface of revolution; 4.4. Area of curvilinear sector; 4.5. Applications in mechanics; P.4. Problems

5: Other Definitions: Riemann and Stieltjes Integrals5.1. Introduction; P.5. Problems; 6: Improper Integrals; P.6. Problems; Part 2: Integration of Several-variable Functions; 7: Additional Properties of Step Functions; 7.1. The notion "almost everywhere"; P.7. Problems; 8: Lebesgue Integral; 8.1. Proof of the correctness of the definition of integral; 8.2. Proof of the Beppo Levi theorem; 8.3. Proof of the Fatou-Lebesgue theorem; P.8. Problems; 9: Fubini and Change-of-Variables Theorems; P.9. Problems; 10: Applications of Multiple Integrals; 10.1. Calculation of the area of a plane figure

10.2. Calculation of the volume of a solid10.3. Calculation of the area of a surface; 10.4. Calculation of the mass of a body; 10.5. The static moment and mass center of a body; 11: Parameter-dependent Integrals; 11.1. Introduction; 11.2. Improper PDIs; P.11. Problems; Part 3: Measure and Integration in a Measure Space; 12: Families of Sets; 12.1. Introduction; P.12. Problems; 13: Measure Spaces; P.13. Problems; 14: Extension of Measure; P.14. Problems; 15: Lebesgue-Stieltjes Measures on the Real Line and Distribution Functions; P.15 Problems

16: Measurable Mappings and Real Measurable FunctionsP.16. Problems; 17: Convergence Almost Everywhere and Convergence in Measure; P.17. Problems; 18: Integral; P.18. Problems; 19: Product of Two Measure Spaces; P.19. Problems; Bibliography; Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

Description based on print version record.

ISBN:

9781119037491

1119037492

9781119037514

1119037514

9781119037385

1119037387

OCLC:

892800065