Based on the authors' combined 35 years of experience in teaching, A Basic Course in Real Analysis introduces students to the aspects of real analysis in a friendly way. The authors offer insights into the way a typical mathematician works observing patterns, conducting experiments by means of looking at or creating examples, trying to understand the underlying principles, and coming up with guesses or conjectures and then proving them rigorously based on his or her explorations. With more than 100 pictures, the book creates interest in real analysis by encouraging students to think geometrically. Each difficult proof is prefaced by a strategy and explanation of how the strategy is translated into rigorous and precise proofs. The authors then explain the mystery and role of inequalities in analysis to train students to arrive at estimates that will be useful for proofs. They highlight the role of the least upper bound property of real numbers, which underlies all crucial results in real analysis. In addition, the book demonstrates analysis as a qualitative as well as quantitative study of functions, exposing students to arguments that fall under hard analysis. Although there are many books available on this subject, students often find it difficult to learn the essence of analysis on their own or after going through a course on real analysis. Written in a conversational tone, this book explains the hows and whys of real analysis and provides guidance that makes readers think at every stage.

A Foundation Course in Mathematics is based on a course on Foundations in one of the most famous and successful mathematics training programme titled Mathematics Training and Talent Search (MTTS) Programme which has been running in India since 1993. The book written in conversational style aims to impart a critical and analytical thinking which will be beneficial for students of any discipline. It also gives emphasis on problem solving and proof writing skills, key aspects of learning mathematics. The Subject matter of the book deals with logic, sets, functions and relations, the essential building blocks for higher mathematics. Numerous problems and exercises are chosen and interspersed inside the sections so that the student participates actively in the discussions. We believe that the study of this book will lay the foundation which will enable the readers to undertake courses in higher mathematics with confidence and due rigour, as has been seen from our experience.

This book, taking a holistic view of geometry, introduces the reader to axiomatic, algebraic, analytic and differential geometry. Starting with an informal introduction to non-Euclidean plane geometries, the book develops the theory to put them on a rigorous footing. It may be considered as an explication of the Kleinian view of geometry a la Erlangen Programme. The treatment in the book, however, goes beyond the Kleinian view of geometry. Some noteworthy topics presented include: various results about triangles (including results on areas of geodesic triangles) in Euclidean, hyperbolic, and spherical planes affine and projective classification of conics twopoint homogeneity of the three planes and the fact that the set of distance preserving maps (isometries) are essentially the same as the set of lengths preserving maps of these planes. Geometric intuition is emphasized throughout the book. Figures are included wherever needed. The book has several exercises varying from computational problems to investigative or explorative open questions.

This book arose out of courses taught by the author. It covers the traditional topics of differential manifolds, tensor fields, Lie groups, integration on manifolds and basic differential and Riemannian geometry. The author emphasizes geometric concepts, giving the reader a working knowledge of the topic. Motivations are given, exercises are included, and illuminating nontrivial examples are discussed. Important features include the following: Geometric and conceptual treatment of differential calculus with a wealth of nontrivial examples. A thorough discussion of the much-used result on the existence, uniqueness, and smooth dependence of solutions of ODEs. Careful introduction of the concept of tangent spaces to a manifold. Early and simultaneous treatment of Lie groups and related concepts. A motivated and highly geometric proof of the Frobenius theorem. A constant reconciliation with the classical treatment and the modern approach. Simple proofs of the hairy-ball theorem and Brouwer's fixed point theorem. Construction of manifolds of constant curvature a la Chern. This text would be suitable for use as a graduate-level introduction to basic differential and Riemannian geometry.