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

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. A Supplementary Text For Undergraduate Courses In The Calculus Of Variations Which Provides An Introduction To Modern Techniques In The Field Based On Measure Theoretic Geometry. Varifold Geometry Is Presented Through And Appraisal Of Plateau's Problem. Functions On Euclidean Space -- Differentiation -- Integration -- Integration On Chains -- Integration On Manifolds. Michael Spivak. Advanced Book Program. Includes Bibliographical References (p. 139-140) And Index. This book uses elementary versions of modern methods found in sophisticated mathematics to discuss portions of "advanced calculus" in which the subtlety of the concepts and methods makes rigor difficult to attain at an elementary level.