…modern non-standard analysis, minus rigor. It is also the way Newton and Leibniz thought about calculus, and the way most physicists intuitively think about problems. For a more mature audience, I like Infinitesimal Calculus by Henle & Kleinberg.
…书中许多结论大多是非标准证明。我非常喜欢它，对这方面感兴趣的人都应该去搞份 PDF 看看。另一本篇幅不长、难度也算入门级的书是詹姆斯·亨勒（James Henle）和尤金·克莱因伯格（Eugene Kleinberg）合著的《微积分入门：基于无穷小的方法》（Infinitesimal Calculus），同样由 Dover 出版［Dover 也出版了凯斯勒（Keisler）教材的纸质版，总的来说是一家很棒的出版社］。当年凯斯勒的书刚出时，有人…
…mostly nonstandard proofs of many of the results in the textbook. I like it a lot, and anyone who's interested in this stuff should grab the PDF. Another book that is short and moderately elementary is "Infinitesimal Calculus" by James Henle and Eugene Kleinberg, also published by Dover [which also publishes a paper version of Keisler's text, and is generally a great publishing company]. When Keisler's book came out, someone…
还有一本我觉得挺有用的书，至少它对超实数系统（hyperreal number system）的介绍很棒：http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mat... 篇幅不长，重点明确。
One more book I found to be useful, at least for its introduction to the hyperreal number system: http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mat... It's short and focused.
While I haven't examined it closely, there's a well thought of, concise, no-fluff book on it: http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mat...
…Robinson's "Non-standard Analysis" and Jerome Keisler's (now free) calculus book someone else mentioned (and the supplement which provides a lot of the theory), people who are interested in this can check out "Infinitesimal Calculus" by James Henle and Eugene Kleinberg (https://store.doverpublications.com/0486428869.html - a short introduction) or "Applied Nonstandard Analysis" by Martin Davis (https://store.doverpublications.com/0486442292.html…
…概率论（Probability Theory，Hoel 著）* The Little Schemer（Friedman 著）。然后再进阶到更深入的教材，比如：* 朴素集合论（Naive Set Theory，Halmos 著）* 线性代数应该这样学（Linear Algebra Done Right，Axler 著）* 几何学新探（Geometry Revisited，Coxeter 著）* 微积分：基于无穷小的方法（Infinitesimal Calculus，Keisler 著）* 具体数学（Concrete Mathematics，Graham 著）* 信息论、推断与学习算法（Information Theory, Inference and Learning，MacKay 著）* SICP（Abelson 著）
…Probability Theory (Hoel) * The Little Schemer (Friedman) The proceed to more advanced texts like: * Naive Set Theory (Halmos) * Linear Algebra Done Right (Axler) * Geometry Revisited (Coxeter) * Infinitesimal Calculus (Keisler) * Concrete Mathematics (Graham) * Information Theory, Inference and Learning (MacKay) * SICP (Abelson)
…现代非标准分析（non-standard analysis），只是没那么严谨。这也是牛顿和莱布尼茨当年思考微积分的方式，以及大多数物理学家凭直觉思考问题的方式。对于更成熟的读者，我推荐亨勒（Henle）和克莱因伯格（Kleinberg）写的《微积分入门：基于无穷小的方法》（Infinitesimal Calculus）。
…modern non-standard analysis, minus rigor. It is also the way Newton and Leibniz thought about calculus, and the way most physicists intuitively think about problems. For a more mature audience, I like Infinitesimal Calculus by Henle & Kleinberg.
…书中许多结论大多是非标准证明。我非常喜欢它，对这方面感兴趣的人都应该去搞份 PDF 看看。另一本篇幅不长、难度也算入门级的书是詹姆斯·亨勒（James Henle）和尤金·克莱因伯格（Eugene Kleinberg）合著的《微积分入门：基于无穷小的方法》（Infinitesimal Calculus），同样由 Dover 出版［Dover 也出版了凯斯勒（Keisler）教材的纸质版，总的来说是一家很棒的出版社］。当年凯斯勒的书刚出时，有人…
…mostly nonstandard proofs of many of the results in the textbook. I like it a lot, and anyone who's interested in this stuff should grab the PDF. Another book that is short and moderately elementary is "Infinitesimal Calculus" by James Henle and Eugene Kleinberg, also published by Dover [which also publishes a paper version of Keisler's text, and is generally a great publishing company]. When Keisler's book came out, someone…
还有一本我觉得挺有用的书，至少它对超实数系统（hyperreal number system）的介绍很棒：http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mat... 篇幅不长，重点明确。
One more book I found to be useful, at least for its introduction to the hyperreal number system: http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mat... It's short and focused.
虽然我还没仔细研读过，但关于这个主题有一本口碑很好、简明扼要、毫无废话的书：http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mat...
While I haven't examined it closely, there's a well thought of, concise, no-fluff book on it: http://www.amazon.com/Infinitesimal-Calculus-Dover-Books-Mat...
…鲁宾逊（Robinson）的《非标准分析》（Non-standard Analysis）以及前面有人提到的杰罗姆·凯斯勒（Jerome Keisler）那本（现在免费了的）微积分教材（还有提供了大量理论证明的补充教材），对此感兴趣的人可以看看詹姆斯·亨勒和尤金·克莱因伯格写的《微积分入门：基于无穷小的方法》（Infinitesimal Calculus，https://store.doverpublications.com/0486428869.html —— 一本简短的入门导论）或者马丁·戴维斯（Martin Davis）的《应用非标准分析》（Applied Nonstandard Analysis，https://store.doverpublications.com/0486442292.html…
…Robinson's "Non-standard Analysis" and Jerome Keisler's (now free) calculus book someone else mentioned (and the supplement which provides a lot of the theory), people who are interested in this can check out "Infinitesimal Calculus" by James Henle and Eugene Kleinberg (https://store.doverpublications.com/0486428869.html - a short introduction) or "Applied Nonstandard Analysis" by Martin Davis (https://store.doverpublications.com/0486442292.html…
…概率论（Probability Theory，Hoel 著）* The Little Schemer（Friedman 著）。然后再进阶到更深入的教材，比如：* 朴素集合论（Naive Set Theory，Halmos 著）* 线性代数应该这样学（Linear Algebra Done Right，Axler 著）* 几何学新探（Geometry Revisited，Coxeter 著）* 微积分：基于无穷小的方法（Infinitesimal Calculus，Keisler 著）* 具体数学（Concrete Mathematics，Graham 著）* 信息论、推断与学习算法（Information Theory, Inference and Learning，MacKay 著）* SICP（Abelson 著）
…Probability Theory (Hoel) * The Little Schemer (Friedman) The proceed to more advanced texts like: * Naive Set Theory (Halmos) * Linear Algebra Done Right (Axler) * Geometry Revisited (Coxeter) * Infinitesimal Calculus (Keisler) * Concrete Mathematics (Graham) * Information Theory, Inference and Learning (MacKay) * SICP (Abelson)