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.

 

Table of Contents

 

Functions on Euclidean Space

Norm and inner Product

Subsets of Euclidean Space

Functions and Continuity Differentiation

 

Definitions

Basic Definitions

Basic Theorems

Partial Derivatives

Inverse Functions

Implicit Functions

Notation

 

Integration

Basic Definitions

Measure Zero and Content Zero

Integrable Functions

Fubinis Theorem

Partitions of Unity

Change of Variable

 

Integration on Chains

Algebraic Preliminaries

Fields and Forms

Geometric Preliminaries

The Fundamental Theorem of Calculus

 

Integration on Manifolds

Manifolds

Fields and Forms on Manifolds

Stokes Theorem on Manifolds

The Volume Element

The Classical Theorems

 

 

Bibliography

Index