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Infinitesimal Calculus review ✓ 0

review Infinitesimal Calculus

Infinitesimal Calculus review ✓ 0 ï ❮BOOKS❯ ✭ Infinitesimal Calculus Author James M. Henle – Rigorous undergraduate treatment introduces calculus at the basic level using infinitesimals and concentrating on theory rather than applications Reuires only a solid foundation in high school mathema School mathematics Contents Introduction Language and Structure The Hyperreal Numbers The Hyperreal Line Continuous Functions Integral CalculusDifferential Calculus The Fundamental Theor. 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 rigorous basis laterThe 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 rigorous explanation is needed the exposition of the ultrafilter construction is amazingly clear while still skipping all time intensive detailsI 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 interested in applications will get an amazingly fast kick start

James M. Henle ñ 0 characters

Em Infinite Seuences and Series Infinite Polynomials The Topology of the Real Line Standard Calculus and Seuences of Functions Appendixes Subject Index Name Index Numerous figures editio. The book is awesome overall but for whomFor freshmen who know nothing about Calculus this book won't do much good for you since the book focuses heavily on the theory not the actual computation If your professor asks you to find a volume of certain solid you'd better look for other conventional textbooksFor those who want to study nonstandard analysis seriously this book will likely be lacking Many proofs depend on one's intuition rather than rigor so don't expect the level of formality you would see from RudinThat being said this book is certainly an awesome primer for nonstandard analysis After reading this I was able to gain intuitive and fairly rigorous understanding of infinitesimals and it was pleasing to prove some important theorems in real analysis in a new light although it could have used rigor as I have written aboveSome Dover books are crazy difficult despite their friendly names Thankfully this book is not the case the book reads easily and fast and is uite thin too I have only taken applied linear algebra principles of real analysis and math for econ class so my knowledge on math is very modestI recommend this book for math hobbyists or anyone who wants to grasp an intuitive understanding of nonstandard analysis before beginning a serious study on this topic

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Infinitesimal CalculusRigorous undergraduate treatment introduces calculus at the basic level using infinitesimals and concentrating on theory rather than applications Reuires only a solid foundation in high. The calculus was created as many know by Newton and Leibniz Newton's concept of calculus was based on continuity while Leibniz used a conceptual framework based on infinitesimals numbers smaller than any real number but less than zero In the 19th century a rigorous basis was established for Newton's conceptual framework but it became an article of faith that infinitesimals could not be rigorously used as a basis for calculus However in the 20th century a rigorous basis was established for an infinitesimal based treatment of the calculus as a result of Abraham Robinson's nonstandard analysis This involves expanding the real number system to a much larger number system the hyperreal number systemIn the physical sciences it is common to use an intuitive treatment of calculus that includes infinitesimals; however nearly all books on basic calculus avoid them and ignore Robinson's ideas I only know of two exceptions a book by H J Keisler who edited Robinson's papers and this one Each has its advantages and disadvantagesKeisler's book is unfortunately out of print and nearly unobtainable It is a complete textbook of calculus using the approach through nonstandard analysis Its treatment of the hyperreal number system however I find hard to understand By contrast this book has a very much clearer treatment of the hyperreals; I think I finally understand how they are constructed after reading this book But this book is not a complete textbook of calculus It covers the theory and covers it extremely well but does not even attempt to teach how to use calculus Therefore it would not be appropriate as a sole textbook in a calculus class for exampleI have read other work by Henle and it is clear that his forte is explaining unusual number systems He does a great job in this book at what he does I just wish he had added material on how to actually use calculus Unfortunately the reader will have to augment this book by another and since no other in print book that I know of uses this nonstandard analysis based approach there will be a disconnect if anyone tries to combine it with another book