Calculus On Manifolds: a Modern Approach To Classical Theorems Of Advanced Calculus

Description

"This little book is especially concerned with those portions of?advanced calculus? in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approach taken here uses elementary versions of modern methods found in sophisticated mathematics. The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course (one which at least mentions the least upper bound (sup) and greatest lower bound (inf) of a set of real numbers). Beyond this a certain (perhaps latent) rapport with abstract mathematics will be found almost essential."--Provided by publisher.  Read more... Abstract: "This little book is especially concerned with those portions of?advanced calculus? in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level. The approach taken here uses elementary versions of modern methods found in sophisticated mathematics. The formal prerequisites include only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course (one which at least mentions the least upper bound (sup) and greatest lower bound (inf) of a set of real numbers). Beyond this a certain (perhaps latent) rapport with abstract mathematics will be found almost essential."--Provided by publisher Content: Cover Half Title Title Copyright Editors' Foreword Preface Contents 1. Functions on Euclidean Space NORM AND INNER PRODUCT SUBSETS OF EUCLIDEAN SPACE FUNCTIONS AND CONTINUITY 2. Differentiation BASIC DEFINITIONS BASIC THEOREMS PARTIAL DERIVATIVES DERIVATIVES INVERSE FUNCTIONS IMPLICIT FUNCTIONS NOTATION 3. Integration BASIC DEFINITIONS MEASURE ZERO AND CONTENT ZERO INTEGRABLE FUNCTIONS FUBINI'S THEOREM PARTITIONS OF UNITY CHANGE OF VARIABLE 4. Integration on Chains ALGEBRAIC PRELIMINARIES FIELDS AND FORMS GEOMETRIC PRELIMINARIES THE FUNDAMENTAL THEOREM OF CALCULUS5. Integration on Manifolds MANIFOLDS FIELDS AND FORMS ON MANIFOLDS STOKES' THEOREM ON MANIFOLDS THE VOLUME ELEMENT THE CLASSICAL THEOREMS Bibliography Index Addenda