Calculus on Manifolds: A Modern Approach to Classical Theorems of Advanced Calculus

✍ Scribed by Michael Spivak

• Publisher: Westview Press
• Year: 1998
• Tongue: English
• Leaves: 159
• 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

Title......Page 2
Copyright Page......Page 3
Editors' Foreword......Page 4
Preface......Page 6
Contents......Page 10
Norm And Inner Product......Page 14
Subsets Of Euclidean Space......Page 18
Functions And Continuity......Page 24
Basic Definitions......Page 28
Basic Theorems......Page 32
Partial Derivatives......Page 38
Derivatives......Page 43
Inverse Functions......Page 47
Implicit Functions......Page 53
Notation......Page 57
Basic Definitions......Page 59
Measure Zero And Content Zero......Page 63
Integrable Functions......Page 65
Fubini's Theorem......Page 69
Partitions of Unity......Page 76
Change of Variable......Page 80
Algebraic Preliminaries......Page 88
Fields And Forms......Page 99
Geometric Preliminaries......Page 110
The Fundamental Theorem Of Calculus......Page 113
Manifolds......Page 122
Fields And Forms On Manifolds......Page 128
Stokes' Theorem on Manifolds......Page 135
The Volume Element......Page 139
The Classical Theorems......Page 147
Bibliography......Page 152
Index......Page 154