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

✍ Scribed by Michael Spivak

• Publisher: Westview Press;CRC Press
• Year: 2018
• Tongue: English
• Leaves: 161
• Edition: First edition
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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

✦ Subjects

Mathematics