Elementary Calculus: An Infinitesimal Approach (Dover Books on Mathematics) 3rd Edition by H. Jerome Keisler (PDF)

Description

This first-year calculus book is centered around the use of infinitesimals, an approach largely neglected until recently for reasons of mathematical rigor. It exposes students to the intuition that originally led to the calculus, simplifying their grasp of the central concepts of derivatives and integrals. The author also teaches the traditional approach, giving students the benefits of both methods.Chapters 1 through 4 employ infinitesimals to quickly develop the basic concepts of derivatives, continuity, and integrals. Chapter 5 introduces the traditional limit concept, using approximation problems as the motivation. Later chapters develop transcendental functions, series, vectors, partial derivatives, and multiple integrals. The theory differs from traditional courses, but the notation and methods for solving practical problems are the same. The text suggests a variety of applications to both natural and social sciences.

User’s Reviews

Editorial Reviews: About the Author H. Jerome Keisler was a longtime professor at the University of Wisconsin, Madison. He pursued his Ph.D. under the direction of Alfred Tarski at the University of California, Berkeley, and his research included model theory and nonstandard analysis.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐The book is well written but the fonts are too small. The price is fair. They need to make bigger fonts and larger pages and Adding to the price of the book.

⭐Wished I had this text 35 years ago when I first took Calculus. I learned the delta-epsilon limit method, but it was not intuitive, albeit rigorous. Now that the intuitive “infinitesimal” method of non-standard analysis has been proved rigorous (by at least two different approaches) it seems ridiculous that mathematics departments at universities cling to the harder method of teaching. In fact, it strikes me as hubris. It is as if they think, well it was hard for me, so it should be hard for the next generation.I would guess that at least 90% of those who take a first course in Calculus do not go on to become mathematicians. Most are engineers, economists, scientists, computer programers etc. and gain nothing by mathematics professors making the subject harder than it has to be. If, as some of the criticisms state, the delta-epsilon method has advantages in that it is mathematical common sense with implications to approximation, nothing precludes the introduction of limits at a later time to the appropriate audience. And in fact, the “traditional” approach is included later in this text.You can download this book for free, and I did. But I find the printed copy to be well worth the price and just more readable for some reason. Others have talked of the problem with the table of contents and exercise answers. I consider these minor issues that can easily be addressed by visiting the author’s website.If you are about to take a course in Calculus, you’re pretty much are going to have to go with the textbook your instructor chooses. If you are not so constrained, you cannot beat this text.

⭐Long overdue text on Infinitesimal Calculus.I’d like to see further school texts and physics texts using the infinitesimal approach.

⭐The book is too large, the author should have spent more time on the fundamentals of hyperreals (he does have a companion book which does treat hyperreals but it’s expensive.)

⭐If you’re looking for an alternative approach to calculus you should try this book. Leibniz, Newton and Robinson are your guide to this interesting world of infinitesimals. This makes calculus more intuitive (although limits are not forgotten in the book).

⭐For those who plan to buy the hard copy of this 3rd edition at this time, you should know that its Table of Contents is INCOMPLETE! I see this new (3rd) edition has 14 chapters in it; however, its TOC shows only up to about a half of chapter 11, then finishes right there. So, you might want to wait until Dover publishers fixes the problem before buying a corrected version.

⭐beautiful book,one should teach calculus along this lines.

⭐I love this book.Keisler follows a consistent, easy to follow approach. Sections are short and cover a specific topic. Concepts are described at an intuitive level, but at the same time the author follows a rigorous approach to the subject. (Occasionally a proof may be omitted; however the author is scrupulous in calling it out, and for the most part this is only the case for minor elements.) Throughout the text there is ample examples along with excellent diagrams.I also really enjoy the infinitesimal approach. I find it simpler in general to follow. However, the nomenclature and presentation of calculus has always had references to infinitesimals, so presenting both epsilon delta and infinitesimal is actually clearer.I am currently working through all the problems with answers. On occasion, there are errors in the answers provided. (I’m always grateful to know that text book authors are human.) I am grateful to the author for the free PDF version. I began with that but I also purchased the text when I realized how much I like it. (It’s nice having hard copy, and I hope at least some remuneration makes it back to the author.)For background I have an undergrad in math and graduate degree in EE. I’m just working through the book for fun. I’m comparing the text to my undergrad (Amherst) text book which I found sub-standard.

⭐Si tratta di un testo che copre gli argomenti svolti tipicamente nei corsi di Analisi Matematica 1, nei corsi di studio di Fisica, Matematica e Ingegneria.La particolarità di questo libro è che costruisce l’analisi matematica a partire dal concetto intuitivo di infinitesimo o numero infinitesimale, cioè di numero diverso da zero ma infinitamente piccolo.Non ho la competenza per giudicare il lavoro svolto dal matematico Abraham Robinson, che negli anni ’60, riprendendo l’impostazione di Leibniz e Newton, re-introdusse gli infinitesimi, fondando di fatto quella che oggi si chiama “analisi non-standard”, che è utilizzata in questo testo.Osservo tuttavia che l’esposizione risulta assai più intuitiva e comprensibile di quella usuale (nella quale si fanno acrobazie, a volte difficilmente comprensibili, per evitare gli infinitesimi) e molto, molto più vicina alle applicazioni dell’analisi, come la fisica, l’economia, la demografia, l’epidemiologia, ecc.Lo consiglio sia agli studenti più brillanti, che con questo testo hanno modo di capire i concetti più profondamente, sia agli studenti che hanno difficoltà in analisi matematica, che con questo testo possono capire meglio concetti che usualmente risultano farraginosi.Muy buen libro, llegó en perfectas condiciones y en muy buen tiempo. Recomiendo altamente el libro para los que deseen adentrarse más a fondo en el cálculo.One of the best book I’ve read on calculus, simple to read and full of explanation and exercises, I’m very satisfied.

⭐According to my son, who’s asked to order this book, it matched all his expectations in terms of calculus. As he’s still 15 years old, this has been helping him learning more about calculus and even clarifying some notions on derivatives, etc.He’d recommend it to everyone interested in Mathematics and knowing more than what’s taught in secondary schools (at least in Portuguese secondary schools).

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