Dr. Robert A. Herrmann

Contents of Book 5.

If you would like a smaller font version that may not be as up-to-date as this version, please see the end of this page for the arxiv version.

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The following zip file (331KB) contains the title page, the contents pages and the first 8 chapters. (Latest revision 11:25 AM EDT, 7/31/03.) The contents of these chapters are as follows:

Chapter 1
Ultrafilters, Cofinite Filter, Principle Ultrafilters, Free Ultrafilters

Chapter 2
A Simple Nonstandard Model for Analysis
Equivalence Classes of Sequences, Totally Ordered Field of Equivalence Classes, The Hyper-extension of Sets and Relations, The Standard Object Generator

Chapter 3
Hyper-set Algebra And Infinite and Infinitesimal Numbers
The Behavior of the Hyper and Standard Object Generators; Infinitesimals, The Infinite and Finite Numbers, *-Transform Process, Maximum Ideal

Chapter 4
Basic Sequential Convergence
Bounded Sequences, Convergent Sequences, Accumulation Point, Subsequences, Cauchy Criterion, Nonstandard Characteristics and Examples

Chapter 5
Advanced Sequential Convergence
Double Sequences, Iterated Limits, Upper and Lower Limits, Nonstandard Characteristics and Examples

Chapter 6
Basic Infinite Series
Hyperfinite Summation, Standard Results, Nonstandard Characteristics and Examples

Chapter 7
An Advance Infinite Series Concept
Multiplying of Infinite Series, Nonstandard Characteristics and Examples

Chapter 8
Additional Real Number Properties
Interior, Closure, Cluster, Accumulation, Isolated Points, Boundedness, Compactness, Nonstandard Characteristics

The zip file is ns1.zip

The following zip file (333KB) contains chapters 9-15 and the Appendix, References, and Index. (Latest revision 1100 EDT, 8/1/03.) The contents of these chapters is as follows:

Chapter 9
Basic Continuous Function Concepts
All Notions Generalized to Cluster Points, One-sided Limits and Continuity, Sum, Product and Composition of Continuous Functions, Extreme and Intermediate Value Theorems, Nonstandard Characteristics

Chapter 10
Slightly Advanced Continuous Function Concepts
Nonstandard Analysis and Bolzano's Product Theorem, Inverse Images of Open Sets, Additive Functions, Uniform Continuity, Extensions of Continuous Functions

Chapter 11
Basic Derivative Concepts
Nonstandard Characteristics for Finite and Infinite Derivatives at Cluster Points, The Infinitesimal Differential, The Fundamental Theorem of Differentials, Order Ideals, Basic Theorems, Generalized Mean Value Theorem, L'Hospital's Rule

Chapter 12
Some Advanced Derivative Concepts
Nonstandard Analysis and the nth-Order Increments, nth-Order Ideals, Continuous Differentiability, Uniformly Differentiable, the Darboux Property, and Inverse Function Theorems

Chapter 13
Riemann Integration
The Simple Partition, Fine Partitions, Upper and Lower Sums, Upper and Lower Hyperfinite Sums, The Simple Integral, The Equivalence of The Simple Integral and The Riemann Integral, The Basic Integral Theorems and How The Generalized and Lebesgue Integral Relate to Fine Partitions.

Chapter 14
What Does the Integral Measure?
Additive Functions and The Rectangular Property

Chapter 15
Metric and Normed Linear Spaces

Existence of Free Ultrafilters, Proof of *-Transform Process

Date of original 7/30/03.

The zip file is ns2.zip

A smaller font version can be found at http://arxiv.org/abs/math.GM/031035

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