A Course in Calculus and Real Analysis: Edition 2

Sudhir R. Ghorpade · Balmohan V. Limaye

Nov 2018 · Springer

About this ebook

Calculus is one of the triumphs of the human mind. It emerged from inv- tigations into such basic questions as ?nding areas, lengths and volumes. In the third century B. C. , Archimedes determined the area under the arc of a parabola. In the early seventeenth century, Fermat and Descartes studied the problem of ?nding tangents to curves. But the subject really came to life in the hands of Newton and Leibniz in the late seventeenth century. In part- ular, they showed that the geometric problems of ?nding the areas of planar regions and of ?nding the tangents to plane curves are intimately related to one another. In subsequent decades, the subject developed further through the work of several mathematicians, most notably Euler, Cauchy, Riemann, and Weierstrass. Today,calculus occupies a centralplacein mathematics and is an essential component of undergraduate education. It has an immense number of app- cations both within and outside mathematics. Judged by the sheer variety of the concepts and results it has generated, calculus can be rightly viewed as a fountainhead of ideas and disciplines in mathematics. Real analysis, often called mathematical analysis or simply analysis, may be regarded as a formidable counterpart of calculus. It is a subject where one revisits notionsencountered in calculus, but with greaterrigor and sometimes with greater generality. Nonetheless, the basic objects of study remain the same, namely, real-valued functions of one or several real variables. This book attempts to give a self-contained and rigorous introduction to calculusoffunctionsofonevariable.

About the author

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.