Dr. Robert A. Herrmann

Contents of Book 2

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First Manual Contents

Cover pages, Contents.

Chapter 1 Introduction. 1.1 A Brief History; 1.2 Manual construction.

Chapter 2 . Infinitesimals, Limited And Infinite Numbers. 2.1 Some Notation and Definitions; 2.2 Basic Algebra; 2.3 Euclidean N-spaces; 2.4 The Standard Part Operator; 2.5 A Slight Quandary.

Chapter 3. Some Set Theory, Convergence And Leibniz' Principle. 3.1 Some Set Theory; 3.2 Convergence; 3.3 Constants and More Constants; 3.4 The Leibniz' Principle of *-transfer; 3.5 A Few Simple Applications.

Chapter 4. Some Modeling With The Infinite Numbers. 4.1 Historical Confusion; 4.2 The Internal Definition Principle; 4.3 Hyperfinite Summation; 4.4 Continuity and a Few Examples.

Chapter 5. Standard Rules For Integral Modeling. 5.1 The Riemann Styled Integral; 5.2 The Infinite Sum Theorems; 5.3 Extensions; 5.4 Applications and The Standard Modeling Rules; 5.5 Extensions of the Standard Rules.

Chapter 6. Nonstandard Rules For Integral Modeling. 6.1 Historical Examples; 6.2 The Monadic Environment; 6.3 Simple Applications; 6.4 The Method of Constants; 6.5 The Hyperfinite Method; 6.6 Instruction; 6.7 Realism. Chapter 7. Pure Infinitesimal Integral Modeling. 7.1 Brief Discussion; 7.2 Geometric Elements.

Chapter 8. Refinements For Integral Modeling. 8.1 A Very General Approach; 8.2 The Line Integral; 8.3 Order Ideals and Approximations; 8.4 nth Order Increments; 8.5 Microgeometry - Tangents to Curves; 8.6 Microgeometry - Surface Elements; 8.7 Microgeometry - Other Stuff; 8.9 Gauge Integrals.

File 1. (523K pdf)

Chapter 9. The Beginnings Of Differential Equation Modeling. 9.1 Brief Discussion; 9.2 The Limit; 9.3 Fluxions and Dynamic Geometry; 9.4 Fluxions and Higher Order Infinitesimals; 9.5 What is a Tangent? 9.6 What is an (k-surface) Osculating Plane? 9.7 Curvature.

Chapter 10. The Differential And Physical Modeling. 10.1 Basic Properties; 10.2 Some General Observations; 10.3 Vibrating Membrane; 10.4 Internal Heat Transfer; 10.5 Concluding Remarks.

APPENDICES

For Chapter 2, For Chapter 3, For Chapter 4, For Chapter 5, For Chapter 6, For Chapter 7, For Chapter 8, For Chapter 10.

REFERENCES, SPECIAL SYMBOLS, INDEX

File 2. (599KB pdf)

Second Manual - Elementary Physics - Contents

Cover Page, Contents

Chapter 1 Introduction. 1.1Brief Comments; 1.2 Manual Structure.

Chapter 2. Mechanics. 2.1 Instantaneous Velocity; 2.2 Acceleration; 2.3 Forces and Newton's Law; 2.4 Vectors; 2.5 Energy and Force Fields; 2.6 General Impulse.

Chapter 3. Slightly Less Basic Mechanics. 3.1 Mass; 3.2 Moments and Center of Mass; 3.3 Point Masses; 3.4 Standard Rules.

References, Special Symbols.

Third Manual - Generalized Functions (Distributions)

1. Additional Modeling Concepts; 2. Generalized Functions; 3. Some Abstract Algebra; 4. Functionals on T; 5. Schwarz Generalized Functions; 6. "Continuity;" 7. Pre-generalized functions and S-continuity; 8. Generalized Functions and Beyond.

File 3. (425KB pdf)

Useful Images

These four drawings may be useful for four applications that appear in this book.

File 4. (6KB zip)A smaller font version can be found at http://arxiv.org/abs/math.GM/031243 http://arxiv.org/abs/math.FM/0403303

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