Infinitesimal Calculus

by James M. Henle, Eugene M. Kleinberg

Rigorous undergraduate treatment introduces calculus at the basic level, using infinitesimals and concentrating on theory rather than applications. Requires only a solid foundation in high school mathematics. 1. Introduction. 2. Language and Structure. 3. The Hyperreal Numbers. 4. The Hyperreal Line. 5. Continuous Functions. 6. Integral Calculus. 7. Differential Calculus. 8. The Fundamental Theorem. 9. Infinite Sequences and Series. 10. Infinite Polynomials. 11. The Topology of the Real Line. 12. Standard Calculus and Sequences of Functions. Appendixes. Subject Index. Name Index. Numerous figures. 1979 edition.

Genres: Mathematics, Reference, Science, Textbooks

144 pages, Paperback

First published March 2, 1979

Reviews

[Maurizio Codogno · Rating 3 out of 5]

Per una volta, il titolo di questo libro è pienamente corretto. Quando a scuola e all'università si parla di "calcolo infinitesimale", infatti, gli infinitesimi sono in realtà tenuti ben lontani: dopo che Cauchy sviluppò la teoria degli epsilon e dei delta i matematici furono ben felici di eliminare quei "fantasmi di quantità evanescenti", come le definì il vescovo Berkeley, che permettevano di risolvere i problemi ma non si sapeva vene come. Peccato che epsilon e delta siano tutto meno che evidenti, a differenza degli infinitesimi... La situazione cambiò negli anni '60 del secolo scorso, quando Robinson riuscì a definire gli infinitesimi in maniera formale. Questo testo spiega appunto come si possono estendere i numeri reali aggiungendo gli infinitesimi (ma anche i numeri interi infiniti e tanti altri numeri...) conservando tutte le proprietà usuali e semplificando i teoremi di analisi matematica. Non essendoci un pranzo gratis, avverto subito che l'estensione non è banale, anche perché è più che altro metamatematica: il libro va insomma bene per chi è davvero interessato al tema e voglia mettersi di buzzo buono a studiare. Carina l'idea di avere una colonna principale con il testo e una laterale con digressioni ed esercizi, per alleggerire la lettura.

[Jessada Karnjana · Rating 5 out of 5]

ผมอ่านเล่มนี้แบบคนที่พอรู้แคลคูลัสมาบ้าง สิ่งที่คาดหวังจากมันจึงไม่ใช่แคลคูลัส แต่เป็นวิธีจัดการกับแคลคูลัสด้วย infinitesimal ซึ่งเป็นแนวทางที่ไลบ์นิซใช้คิดแคลคูลัส ขณะที่ความคิดของนิวตันถูกพัฒนาต่อไปเป็นทฤษฎีลิมิต และหลักสูตรเลข ม.ปลาย ของเราก็สอนแคลคูลัสผ่านกรอบความคิดแบบลิมิต เหตุผลหนึ่งคือ ลิมิตถูกพัฒนาให้มีความหนักแน่นในเชิงตรรกะทางคณิตศาสตร์ก่อน อะไรที่เสร็จก่อนและดีใช้ได้ ก็จะได้เป็นมาตรฐาน ส่วน infinitesimal ของไลบ์นิซเพิ่งถูกทำให้หนักแน่นไม่แพ้กันโดยอับราฮัม โรบินสัน เมื่อห้าหกสิบกว่าปีที่แล้ว ทำให้ได้รับป้ายชื่อว่าเป็น nonstandard analysis ไอเดียของไลบ์นิซนั้นเกี่ยวข้องกับจำนวนชนิดหนึ่งที่ไม่ใช่จำนวนจริง และไม่ใช่จำนวนจินตภาพ จำนวนตัวนั้นคือ infinitesimal พูดแบบหยาบ ๆ มันคือจำนวนที่มีค่ามากกว่า 0 แต่น้อยกว่าจำนวนจริงบวกที่มีค่าน้อยที่สุด คำถาม จำนวนจริงบวกที่มีค่าน้อยที่สุดนะมีเหรอ? ...

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[J. · Rating 3 out of 5]

Henle and Kleinberg are really good writers. Things are as clear as they can be, and there seems to be a good sense of humor in various places--certainly something that isn't very common in a math text. (For instance, see the examples of pictorial functions on pages 19-23.) I'm not totally sold on the infinitesimal method--the proofs are all really easy, but (perhaps just because I don't have a good intuition for infinitesimals), I don't feel like I internalized many of them, like maybe they're TOO easy. Having said that, the usual epsilon-delta stuff isn't exactly the world's most intuitive thing, either, so maybe they have a point. Anyway, this is a great intro to the subject, and I'll probably be looking further, so that's as good as you can hope for, yes?

[Salaw · Rating 5 out of 5]

A lovely little book, which does a very nice job of laying out the basics of calculus using the hyperreals. It's also extremely readable, and, in my opinion, it does a pretty good job of carrying Gene Kleinberg's very accessible lecture style onto paper. (I can't comment on Henle's lecture style; though I've read he's great, I've never heard him speak.) My only real objection is that the authors make use of proof by contradiction in some cases where they could have used a more constructive style, but it's a small nit.