Infinitesimal Calculus: And, Hippolytus

?In this book, you will find an easy, and intuitive construction of hyper-reals. the book proceeds to prove basic theorems of analysis using the hyper-reals. I believe that anyone with a high school calculus can understand the main ideas in this book. The book is very easy to follow, and yet the results, and insights from the book are quite relevant.

Many books on calculus using hyperreals focus on the strict formalism. For the hard-core mathematician this might be the adequate approach, however for us normal people with normal questions (e.g. "WTF ARE hyperreals?") this short and concise book appears to be the first choice. Easy reading.

I've since the starting in school 58 ago been using calculation and mathematic and always liked mathematic and historic more than anything else. And concerning the mathematic this is excellent as we have mathematic everywhere. Now during 35 years I've received many Dover books, and always without the possibility at first looking inside these books, neither in the Dover catalogues or on the PC sides when going to Amazon. So frequently I've received books which weren't easy to use for self studying, but the actual book "Infinitesimal Calculus" really is a pearl, or a diamond to use. As side remark I last in the writing have put in some lines concerning Newton and Leibniz that is: "Who invented the Integration and the Differentiation". Because in nearly all books, and thereby also by the teachers, we are learning that Newton and Leibniz independently invented this part in mathematic. Which I after all my readings don't find fitting together. As we in Amazon, or other places, for example in the catalogues from Dover, can't look at the sides in the books, I hereby have written down the titles on the parts in this book, remembering about how it would have helped me if had been there when I was speculation about buying this book: Preface vi 1. Introduction 3 2. Language and Structure 13 3. The Hyperreal Numbers 25 4. The Hyperreal Line 32 5. Continues Functions 42 6. Integral Calculus 52 7. Differential Calculus 65 8. The Fundamental Theorem 77 9. Infinite Sequential and Series 84 10. Infinite Polynomials 95 11. The Topology of the Real Line 106 12. Standard Calculus and Sequences of Functions 116 Appendix A. Defining Quasi-big Sets 126 Appendix B. The Proof of Theorem 3.1 128 Subject Index 130 Name Index 134 It's a delight that through out the book, on left margin on nearly all of the sides, we are having a parallel writing with examples, exercises, proofs or drawings. And for example in chapter 6, with the "Integral Calculus", we again are getting the picture used as the front cover of the book. But the three-dimensional drawing on the front cover actual are the step number 2 in geometrically showings with parallel showing how to get to an actual formula. This drawing, the one on front of the book, is where 2 quarters of a "pyramid" have been put together, after we in the first drawing only are having one of them. And the drawing number 2, we then in the two following drawings have piled up 2 times more. First with one more quarter, and then with a copy of all of these 3 put on the top, after first having been turned upside down. And thereby ending with a bloc existing of 6 quarter pyramids, and now having a box with the sides n, n+1, and 2n+1, and thereby the total volume: n(n + 1)(2n + 1). And as we, as stated, repeated 6 times with the same quarter of pyramids we thereby are getting the Sigma formula for the original "quarter pyramid" to be: . Actually in the start of this chapter we first have got the proof, and 2 drawings, concerning the two-dimensional squares, that is squares, like flags on a floor, and together making a triangle with two equal long sides both having the numbers on n. And finish with showing that we end with the Sigma being: . In the book we are having many drawings actually I get it to be around 135, of course much depending on what we regard as being a drawing, as for example a line with only some remarks put onto it. But in any case, to me, the more geometrical drawings we in the mathematic are getting, the better. I owe science books concerning nearly everything, but most concerning mathematic, actual 400. But in nearly al of the mathematic books, including this one, we are told that independently Newton and Leibniz both invented the Integration and the Differentiation. But during my reading I have come to the conclusion that it only was Newton that invented the Integration and Differentiation. Even though the Chinese actual at the same time made Integration. But to point it out, I'm not born in England, so this is not the reason for my support to Newton. But concerning this case it's a fact that Leibniz some years were working in the German embassy in London. And that in these years, in London, in the science group, there was circulating a paper in which Newton showed his invention of the Integration. And it's also known that Leibniz have been reading these papers, and besides that Newton and Leibniz, during many years were communicating. And then when Leibniz returned back to Germany he wrote his book with the Integration and Differentiating. And as normally, in history, not mentioning anything if the knowledge had come from another person. And as this became the first printed book with Integration and Differentiation, it through out Europe was thought that Leibniz was the inventor. In reality the only thing that Leibniz had done, was that he gave us the nice Integration sign which we are using, and also for the Differentiation gave us the normal used marks telling how many times we want to have the Differentia repeated. But where we in the engineering physic calculations are using the Differentiating signs as Newton was using them. But to put it hard: Leibniz signs are nice, but like a person who now a day has designed a new car, or painted a car, not can say that he has invented the car, neither can Leibniz say that he have invented the Integration and Differentiation! But in history we have enormous many cases with formulas being named after the wrong persons, but in many cases this because the first invention never was know by the second inventor, and where the second inventor then came with it at the right time. But for example Pythagoras triangle formula in India and China was known around 1,000 years before he came with it. But we from out there don't have any personal names connected to it, so therefore it's all right to call this formula for Pythagoras triangle. Opposite to the case between Newton and Leibniz. Opposite to Leibniz it was close to that we would have got Robert Hooke for to Newton being a parallel inventor of the Integration. But now a day we nearly only knows about him from the Hooks Law (concerning bending), because he became an enemy to Newton.

This is a great book, especially if ... you need to learn the important proofs and structure of calculus in a limited amount of time (a basic calculus curriculum covered in 120 small pages!), or ... you want to get an intuitive feel for the hyperreals to be able to apply them in basic cases or learn a more rigorous basis later. The authors do a great job at keeping things simple. They made wise choices on when an intuitive grasp of a concept is sufficient (such as extension of functions to the hyperreals) and when a more rigorous explanation is needed (the exposition of the ultrafilter construction is amazingly clear, while still skipping all time-intensive details). I really appreciate that the authors had the bravery to skip a lot of technical details - something that is rarely seen in mathematics, even in introductory texts where doing so would make sense. Of course, in skipping things for the sake of seeing the big picture, at times you may find situations where you are forced to look for additional information from other sources. But this just opens a path for those interested in the deeper math behind the hyperreals, while those more interested in applications will get an amazingly fast kick-start.

I got this book from the library expecting to get a brief review of the Calculus. After reading the text I ordered it right away. This is an exceptionally beautiful book that gives one a rigorous understanding of real analysis. During my read I missed fully understanding portions of the text so I had to read some parts more than once. Hence it took me about a month to finish the 144 page book amongst other readings. If you are looking for a great Calc book with a goodly amount of rigor and you have the time and inclination to keep at it this book will not let you down. What amazes me is how much of import and beauty is cramped within the 144 pages of this text. In comparing this book to many of the huge over-priced, glitzy Calculus textbooks; so prevalent today, this book gives one a ton of return for the money and the effort expended in understanding its contents. Dover Books has done us a great service once again by making such a valuable and excellent work so affordable and in a great presentation format with regards to font size, diagrams etc. What a tremendous bargain. Definitely a 'must have'.