Stroyan K. D. . (1997). Mathematical Background: Foundations of Infinitesimal Calculus second edition. Academic Press, Inc.

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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 explanations. The projects bring out these issues in the way we and best for most students, but the pure mathematical questions also interest some students. This book of mathematical \background" shows how to all 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 de�nition is WRONG in the sense that it does NOT support the intended idea. You might agree that the counterintuitive consequences of pointwise derivatives are unfortunate, 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" derivatives. It actually is not possible to give a formula that is pointwise di�erentiable and not uniformly di�erentiable. 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 de�ned. 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 comparison. 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 �nd in the literature. It includes a complete self-contained treatment of Robinson's modern theory of in�nitesimals, �rst discovered in 1961. Our simple treatment is due to H. Jerome Keisler from the 1970's. Keisler's elementary calculus using in�nitesimals is sadly out of print. It used pointwise derivatives, but had many novel ideas, including the �rst modern use of a microscope to describe the derivative. (The l'Hospital/Bernoulli calculus text of 1696 said curves consist of in�nitesimal straight segments, but I do not know if that was associated with a magnifying transformation.) In�nitesimals give us a very simple way to understand the uniform 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 �nd in traditional calculus sources. The microscope theorem for di�erential 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 proof.

Jenis Artikel: Buku
Tema: Matematika
Program Studi: Fakultas Keguruan dan Ilmu Pendidikan > S1 Pendidikan Matematika
Depositing User: Unnamed user with email [email protected]
Date Deposited: 04 Mar 2019 02:54
Last Modified: 05 Mar 2019 07:20
URI: http://eprints.umbjm.ac.id/id/eprint/764

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