A Course in Calculus and Real Analysis
Authors: Ghorpade, Sudhir R., Limaye, Balmohan V.
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 About this Textbook

This book provides a selfcontained 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.
 About the authors

 Reviews

“Ghorpade and Limaye (both, Indian Institute of technology, Bombay) have written a very coherent and rigorous advanced calculus text on functions of one variable. … Theorems are carefully proven and exercises (without solutions are provided. … Recommended. Upperdivision undergraduates.” ( R. L. Pour, Choice, February, 2018)
“A Course in Calculus and Real Analysis by S.R. Ghorpade and B.V. Limaye (Springer UTM) is a rigorous, wellpresented and original introduction to the core of undergraduate mathematics — firstyear calculus. … This book is a tour de force, and a necessary addition to the library of anyone involved in teaching calculus, or studying it seriously.” (N.J. Wildberger, Gazette of the Australian Mathematical Society, Vol. 35 (3), 2008)
From the reviews:
"The book is a selfcontained and rigorous introduction to calculus of one variable. It covers the programme of the firstyear in mathematical analysis. The main goal of the book is a clear and successful presentation of the basic ideas of analysis. The reader can see an attention of the authors to help students in understanding and further developing of the basic notions of the subject. … are illustrated by a number of interesting examples and remarks. In some sense it is a book for selfeducation." (Sergi V. Rogosin, Zentralblatt MATH, Vol. 1100 (2), 2007)
 Table of contents (9 chapters)


Numbers and Functions
Pages 142

Sequences
Pages 4366

Continuity and Limits
Pages 67102

Differentiation
Pages 103146

Applications of Differentiation
Pages 147178

Table of contents (9 chapters)
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Bibliographic Information
 Bibliographic Information

 Book Title
 A Course in Calculus and Real Analysis
 Authors

 Sudhir R. Ghorpade
 Balmohan V. Limaye
 Series Title
 Undergraduate Texts in Mathematics
 Copyright
 2006
 Publisher
 SpringerVerlag New York
 Copyright Holder
 SpringerVerlag New York
 eBook ISBN
 9780387364254
 DOI
 10.1007/0387364250
 Series ISSN
 01726056
 Edition Number
 1
 Number of Pages
 X, 432
 Topics