Sudhir R. Ghorpade, Balmohan V. Limaye

A Course in Calculus and Real Analysis (eBook, PDF)

• Format: PDF

Produktbeschreibung

This book provides a self-contained and rigorous introduction to calculus of functions of one variable. The presentation and sequencing of topics emphasizes the structural development of calculus. At the same time, due importance is given to computational techniques and applications. The authors have strived to make a distinction between the intrinsic definition of a geometric notion and its analytic characterization. Throughout the book, the authors highlight the fact that calculus provides a firm foundation to several concepts and results that are generally encountered in high school and accepted on faith. For example, one can find here a proof of the classical result that the ratio of the circumference of a circle to its diameter is the same for all circles. Also, this book helps students get a clear understanding of the concept of an angle and the definitions of the logarithmic, exponential and trigonometric functions together with a proof of the fact that these are not algebraic functions. A number of topics that may have been inadequately covered in calculus courses and glossed over in real analysis courses are treated here in considerable detail. As such, this book provides a unified exposition of calculus and real analysis.

The only prerequisites for reading this book are topics that are normally covered in high school; however, the reader is expected to possess some mathematical maturity and an ability to understand and appreciate proofs. This book can be used as a textbook for a serious undergraduate course in calculus, while parts of the book can be used for advanced undergraduate and graduate courses in real analysis. Each chapter contains several examples and a large selection of exercises, as well as "Notes and Comments" describing salient features of the exposition, related developments and references to relevant literature.

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Produktdetails

• Verlag: Springer-Verlag GmbH
• Seitenzahl: 432
• Erscheinungstermin: 14. Oktober 2006
• Englisch
• ISBN-13: 9780387364254
• Artikelnr.: 38208315

Autorenporträt

Sudhir R. Ghorpade is Institute Chair Professor in the Department of Mathematics at the Indian Institute of Technology (IIT) Bombay. He has received several awards, including the All India Council for Technical Education (AICTE) Career Award for Young Teachers and the Prof. S.C. Bhattacharya Award for Excellence in Pure Sciences. His research interests lie in algebraic geometry, combinatorics, coding theory, and commutative algebra. Balmohan V. Limaye is Professor Emeritus in the Department of Mathematics at the Indian Institute of Technology (IIT) Bombay. He is the author of several research monographs and textbooks, including Linear Functional Analysis for Scientists and Engineers (Springer, 2016). He worked at IIT Bombay for more than 40 years and has twice received the Award for Excellence in Teaching from IIT Bombay. His research interests include Banach algebras, approximation theory, numerical functional analysis, and linear algebra. The authors' companion volume A Course in Multivariable Calculus and Analysis (2010) is also in the UTM series.

Inhaltsangabe

1. Numbers and Functions.- 2. Sequences.- 3. Continuity and Limits.- 4. Differentiation.- 5. Applications of Differentiation.- 6. Integration.- 7. Elementary Transcendental Functions.- 8. Applications and Approximations of Riemann Integrals.- 9. Infinite Series and Improper Integrals.- 10. Sequences and Series of Functions, Integrals Depending on a Parameter.- A. Construction of the Real Numbers.- B. Fundamental Theorem of Algebra.- References.- List of Symbols and Abbreviations.- Index.

Rezensionen

"This book would be a valuable asset to a university library and that many instructors would do well to have a copy of this book in their personal libraries. In addition, I believe that some students would benefit if they possessed a copy of this book to use for reference purposes." (Jonathan Lewin, MAA Reviews, April 15, 2019)