Robinsons modern infinitesimal approach puts the intuitive ideas of the. While this book is the go to resource for infinitesimal calculus if you are interested in gaining a basic understanding of its construction I would recommend 'Infinitesimal calculus' by Henle which provides an approachable construction of the hyppereal's and proof of the transfer theorem there is also 'Foundations of infinitesimal calculus' by Keisler which covers Standardisation and slightly more rigorous proofs for calculus, both single and multivariate (including vector calculus). This is a calculus textbook at the college Freshman level based on Abraham. like limit calculus books you not learn higher analysis, using mathematical logic, constructing the number system and such, you will learn only the tools you need but like the use of limits, infinitesimal calculus's tools make sense, in that you don't need higher analysis to justify to yourself using it or to make sense of it. Elementary Calculus: An Infinitesimal approach is a textbook by H. You will not learn about the construction of the hyperreal numbers, or Model theory, you will (like with limit calculus) learn the tools for using calculus, in this case transfer and standardisation. This book provides a very clear and intuitive description of infinitesimal calculus, which falls within the domain of Non-standard analysis which, by the way, has been shown to be logically equivalent to Real analysis. Finding this book was vindicating in that respect.
I never understood why infinitesimals were not considered rigorous, even rough handling of them leads to them to the same conclusions as limit calculus, it couldn't just be a coincidence that they work.
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