Preface

This is the text for a two-semester multivariable calculus course. The setting is n-dimensional Euclidean space, with the material on differentiation culminating in the Inverse Function Theorem and its consequences, and the materialon integration culminating in the Generalized Fundamental Theorem of Inte-gral Calculus (often called Stokes’s Theorem) and some of its consequences inturn. The prerequisite is a proof-based course in one-variable calculus. Some familiarity with the complex number system and complex mappings is occasionally assumed as well, but the reader can get by without it.The book’s aim is to use multivariable calculus to teach mathematics asa blend of reasoning, computing, and problem-solving, doing justice to thestructure, the details, and the scope of the ideas.

To this end, I have triedto write in a style that communicates intent early in the discussion of eachtopic rather than proceeding coyly from opaque definitions.Also, I have triedoccasionally to speak to the pedagogy of mathematics and itseffect on theprocess of learning the subject. Most importantly, I have tried to spread theweight of exposition among figures, formulas, and words. Thepremise is thatthe reader is ready to do mathematics resourcefully by marshaling the skillsof•geometric intuition (the visual cortex being quickly instinctive),•algebraic manipulation (symbol-patterns being precise and robust),•and incisive use of natural language (slogans that encapsulate central ideasenabling a large-scale grasp of the subject).

Thinking in these ways renders mathematics coherent, inevitable, and fluent.In my own student days I learned this material from books by Apostol,Buck, Rudin, and Spivak, books that thrilled me. My debt to those sourcespervades these pages. There are many other fine books on the subject as well,such as the more recent one by Hubbard and Hubbard. Indeed, nothing inthis book is claimed as new, not even its neuroses. Whatever improvementthe exposition here has come to show over the years is due to innumerableideas and comments from my students in turn.