Mathematical Backround Foundations of Infintesimal Calculus.pdf

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Mathematical Background:
Foundations of Infinitesimal Calculus
second edition
by
K. D. Stroyan
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A Microscopic View of the Tangent
Copyright c 1997 by Academic Press, Inc. - All rights reserved.
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Preface to the Mathematical Background
We want you to reason with mathematics. We are not trying to get everyone to give
formalized proofs in the sense of contemporary mathematics; ‘proof’ in this course means
‘convincing argument.’ We expect you to use correct reasoning and to give careful expla-
nations. The projects bring out these issues in the way we find best for most students,
but the pure mathematical questions also interest some students. This book of mathemat-
ical “background” shows how to fill in the mathematical details of the main topics from
the course. These proofs are completely rigorous in the sense of modern mathematics –
technically bulletproof. We wrote this book of foundations in part to provide a convenient
reference for a student who might like to see the “theorem - proof” approach to calculus.
We also wrote it for the interested instructor. In re-thinking the presentation of beginning
calculus, we found that a simpler basis for the theory was both possible and desirable. The
pointwise approach most books give to the theory of derivatives spoils the subject. Clear
simple arguments like the proof of the Fundamental Theorem at the start of Chapter 5 below
are not possible in that approach. The result of the pointwise approach is that instructors
feel they have to either be dishonest with students or disclaim good intuitive approximations.
This is sad because it makes a clear subject seem obscure. It is also unnecessary – by and
large, the intuitive ideas work provided your notion of derivative is strong enough. This
book shows how to bridge the gap between intuition and technical rigor.
A function with a positive derivative ought to be increasing. After all, the slope is
positive and the graph is supposed to look like an increasing straight line. How could the
function NOT be increasing? Pointwise derivatives make this bizarre thing possible - a
positive “derivative” of a non-increasing function. Our conclusion is simple. That definition
is WRONG in the sense that it does NOT support the intended idea.
You might agree that the counterintuitive consequences of pointwise derivatives are un-
fortunate, but are concerned that the traditional approach is more “general.” Part of the
point of this book is to show students and instructors that nothing of interest is lost and a
great deal is gained in the straightforward nature of the proofs based on “uniform” deriva-
tives. It actually is not possible to give a
formula
that is pointwise differentiable and not
uniformly differentiable. The pieced together pointwise counterexamples seem contrived
and out-of-place in a course where students are learning valuable new rules. It is a theorem
that derivatives computed by rules are automatically continuous where defined. We want
the course development to emphasize good intuition and positive results. This background
shows that the approach is sound.
This book also shows how the pathologies arise in the traditional approach – we left
pointwise pathology out of the main text, but present it here for the curious and for com-
parison. Perhaps only math majors ever need to know about these sorts of examples, but
they are fun in a negative sort of way.
This book also has several theoretical topics that are hard to find in the literature. It
includes a complete self-contained treatment of Robinson’s modern theory of infinitesimals,
first discovered in 1961. Our simple treatment is due to H. Jerome Keisler from the 1970’s.
Keisler’s elementary calculus using infinitesimals is sadly out of print. It used pointwise
derivatives, but had many novel ideas, including the first modern use of a microscope to
describe the derivative. (The l’Hospital/Bernoulli calculus text of 1696 said curves consist
of infinitesimal straight segments, but I do not know if that was associated with a magni-
fying transformation.) Infinitesimals give us a very simple way to understand the uniform
ii
derivatives, although this can also be clearly understood using function limits as in the text
by Lax, et al, from the 1970s. Modern graphical computing can also help us “see” graphs
converge as stressed in our main materials and in the interesting Uhl, Porta, Davis,
Calculus
& Mathematica
text.
Almost all the theorems in this book are well-known old results of a carefully studied
subject. The well-known ones are more important than the few novel aspects of the book.
However, some details like the converse of Taylor’s theorem – both continuous and discrete –
are not so easy to find in traditional calculus sources. The microscope theorem for differential
equations does not appear in the literature as far as we know, though it is similar to research
work of Francine and Marc Diener from the 1980s.
We conclude the book with convergence results for Fourier series. While there is nothing
novel in our approach, these results have been lost from contemporary calculus and deserve
to be part of it. Our development follows Courant’s calculus of the 1930s giving wonderful
results of Dirichlet’s era in the 1830s that clearly settle some of the convergence mysteries
of Euler from the 1730s. This theory and our development throughout is usually easy to
apply. “Clean” theory should be the servant of intuition – building on it and making it
stronger and clearer.
There is more that is novel about this “book.” It is free and it is not a “book” since it is
not printed. Thanks to small marginal cost, our publisher agreed to include this electronic
text on CD at no extra cost. We also plan to distribute it over the world wide web. We
hope our fresh look at the foundations of calculus will stimulate your interest. Decide for
yourself what’s the best way to understand this wonderful subject. Give your own proofs.
Contents
Part 1
Numbers and Functions
Chapter 1. Numbers
1.1 Field Axioms
1.2 Order Axioms
1.3 The Completeness Axiom
1.4 Small, Medium and Large Numbers
Chapter 2. Functional Identities
2.1 Specific Functional Identities
2.2 General Functional Identities
2.3 The Function Extension Axiom
2.4 Additive Functions
2.5 The Motion of a Pendulum
Part 2
Limits
Chapter 3. The Theory of Limits
3.1 Plain Limits
3.2 Function Limits
3.3 Computation of Limits
Chapter 4. Continuous Functions
4.1 Uniform Continuity
4.2 The Extreme Value Theorem
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Contents
4.3 Bolzano’s Intermediate Value Theorem
Part 3
1 Variable Differentiation
Chapter 5. The Theory of Derivatives
5.1 The Fundamental Theorem: Part 1
5.1.1 Rigorous Infinitesimal Justification
5.1.2 Rigorous Limit Justification
5.2 Derivatives, Epsilons and Deltas
5.3 Smoothness
⇒
Continuity of Function and Derivative
5.4 Rules
⇒
Smoothness
5.5 The Increment and Increasing
5.6 Inverse Functions and Derivatives
Chapter 6. Pointwise Derivatives
6.1 Pointwise Limits
6.2 Pointwise Derivatives
6.3 Pointwise Derivatives Aren’t Enough for Inverses
Chapter 7. The Mean Value Theorem
7.1 The Mean Value Theorem
7.2 Darboux’s Theorem
7.3 Continuous Pointwise Derivatives are Uniform
Chapter 8. Higher Order Derivatives
8.1 Taylor’s Formula and Bending
8.2 Symmetric Differences and Taylor’s Formula
8.3 Approximation of Second Derivatives
8.4 The General Taylor Small Oh Formula
8.4.1 The Converse of Taylor’s Theorem
8.5 Direct Interpretation of Higher Order Derivatives
8.5.1 Basic Theory of Interpolation
8.5.2 Interpolation where
f
is Smooth
8.5.3 Smoothness From Differences
Part 4
Integration
Chapter 9. Basic Theory of the Definite Integral
9.1 Existence of the Integral
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