" I do. ... " Notices of the AMS, August 1999 "... This book is a pleasure to hold and to look at: ample margins, nice photos, instructive pictures and beautiful drawings .

The book is arranged so that each chapter builds upon the other, giving students a gradual understanding of the subject. This text is part of the Walter Rudin Student Series in Advanced Mathematics.

This is then collated in the last chapter to present Chern's proof of the Gauss-Bonnet theorem for compact surfaces.

Skillfully organized introductory text examines origin of differential equations, then defines basic terms and outlines the general solution of a differential equation.

Review from the first edition: "This book is intended for the student who has a good, but naïve, understanding of elementary calculus and now wishes to gain a thorough understanding of a few basic concepts in analysis.

This first-year calculus book is centered around the use of infinitesimals.

In addition to the standard results of advanced calculus, the book contains several interesting applications of these results. The text is intended to form a bridge between calculus and analysis.

A list of updates is found in the Preface to this edition. This text is based on the author’s experience in teaching graduate courses and the minimal requirements for successful graduate study.

This new edition has been revised by the author, to include several new sections and a new appendix.

The second half of this book, and consequently the second semester, covers differentiation and integration, as well as the connection between these concepts, as displayed in the general theorem of Stokes.