Analysis and beyond an introduction with examples and exercises / [electronic resource] :
Shigeru Kanemitsu, Takako Kuzumaki, Jianya Liu.
- Singapore : World Scientific, 2021.
- 1 online resource (xiii, 313 p.)
Includes bibliographical references and index.
Preface -- Preliminaries -- Continuity of real numbers -- Limits and continuity -- Properties of continuous mappings -- Differentiation -- Properties of differentiable functions -- Extremal values, mean value theorems and taylor expansions in one variable -- Extremal values, mean value theorems and Taylor expansion in several variables -- Algorithms -- Theory of Riemann-Stieltjes integration -- Theory of Riemann integration -- Limit values and summability -- ODE and FDE -- Vector analysis -- Theory of distributions -- Topological spaces -- Appendices. Basics from linear algebra. Diagonalization of matrices. Circulants. Formulas for vector operations. Alternating algebras -- Bibliography -- Index.
"This volume aims to bridge between elementary textbooks on calculus and established books on advanced analysis. It provides elucidation of the reversible process of differentiation and integration through two featured principles: the chain rule and its inverse - the change of variable - as well as the Leibniz rule and its inverse - the integration by parts. The chain rule or differentiation of composite functions is ubiquitous since almost all (a.a.) functions are composite functions of (elementary) functions and with the change of variable method as its reverse process. The Leibniz rule or differentiation of the product of two functions is essential since it makes differentiation nonlinear and with the method of integration by parts as its reverse process. Readers will find numerous worked-out examples and exercises in this volume. Detailed solutions are provided for most of the common exercises so that readers remain enthusiastically motivated in solving and understanding the concepts better. The intention of this volume is to lead the reader into the rich fields of advanced analysis and to obtain a much better view of useful mathematics"--Publisher's website.
Mode of access: World Wide Web. System requirements: Adobe Acrobat Reader.