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I am currently working on a paper to be submitted to a US journal (addressed primarily to non-mathematicians’ audience) where I use

(1) some standard calculus stuff (e.g. limits, Taylor expansions, integration by parts) and (2) some standard probability theory facts (e.g. Central Limit Theorem, Chebyshev’s inequality).

What textbooks would you advise me to list as references for these topics so that the readers could find these topics covered there? I am looking for books that are well known in the US (and not hard to access), contain full proofs but are not too hard for non-mathematicians to comprehend? Thank you.

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    $\begingroup$ Off the top of my head, it is hard for me to imagine an adult reader who does not know standard calculus material but would be willing to learn it in order to read any single paper. Or am I wrong about this? Anyway, for a reference I would suggest something which is freely available on the internet. Is there some reason why "See wikipedia, e.g. the following articles...?" would not be sufficient? Note that a lot of mathematically minded people have worked pretty hard on wikipedia's basic math articles over the last five years (including me): it's really quite good now. $\endgroup$ Commented Apr 29, 2011 at 18:56
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    $\begingroup$ And I hope you list them by page number, not merely a 1000-page textbook... $\endgroup$ Commented Apr 29, 2011 at 19:08
  • $\begingroup$ If the article is for people who may not even know calculus, I would just put all proofs and involved calculations in an appendix. I don't think you need to recommend reference books, but you probably should try to name all concepts correctly and explicitely, so that the reader can look them up ("From (7) we get (8) by integration by parts."). $\endgroup$ Commented Jan 27, 2014 at 9:16

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I think Gilbert Strang's Calculus not only has all the calculus and probability the average (and not so average) beginner needs - done carefully but highly intuitively with lots of pictures - but best of all, it's available online for free.

Can't get better than that for any recommendation for a beginning calculus student.

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Tom Apostol's Calculus is a "calculus" textbook with proofs and contains two chapters on probability. But then again, such isn't exactly a textbook for "non-mathematicians". Even though the text is suitable for students with good high-school mathematics background, it seems unlikely that anyone without a serious interest in mathematics would be willing to sit through over thousand pages of...well, proofs. It is my understanding that probability textbooks "with proofs" tend to assume, at the very least, multivariable calculus as a prerequisite.

Also, I am not sure if there are calculus textbooks "with proofs" in the current market that aren't written for serious students of mathematics.

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For standard calculus stuff, "A course of pure mathematics" by Hardy seems to fit what you want pretty well.

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    $\begingroup$ @Snark: I happen to have acquired a copy of this book recently, and I disagree with you. Hardy's book is elementary, but serious. On the occasion of the centenary edition, Tom Korner wrote a very thoughtful preface explaining that we do not teach calculus / analysis in this way anymore. We give a much less serious version to non-majors, and serious math majors are led quickly to much more sophisticated topics like metric spaces and measure theory. I don't see what a non-mathematician would get out of Hardy's book that they wouldn't get out of Stewart's calculus, or better, wikipedia. $\endgroup$ Commented Apr 29, 2011 at 21:14
  • $\begingroup$ The list of calculus stuff in the question is "limits, Taylor expansions, integration by parts"... Metric spaces and measure theory definitely don't fit. The wikipedia entries only give the bare minimum. There is a middle ground of mathematics between "just know how to work with things" and "know the advanced stuff" that this book covers conveniently in my opinion. $\endgroup$ Commented Apr 30, 2011 at 9:12
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    $\begingroup$ @Snark: Have you read Korner's preface to the centenary edition? I highly recommend it. His point is that the curriculum in England (and his remarks apply very well to the US as well) has changed a lot since Hardy's book was written: its combination of treating relatively elementary topics with a large amount of rigor and expecting that the reader can fill in arguments on her own is pretty far from where we are nowadays. My point was that an educated adult in 2011 who has never taken calculus could hardly have any experience in mathematics whatsoever... $\endgroup$ Commented Apr 30, 2011 at 18:15
  • $\begingroup$ ...(and, I suspect, relatively little interest), so would find this book prohibitively hard to read. Let me put it this way: I am (like you) a researcher in pure mathematics. I very much enjoy Hardy's book AND AM LEARNING FROM IT MYSELF. I have taught many calculus classes, so I know that this book would be appropriate for very few beginning calculus students. Moreover, the wikipedia entries actually have a lot detail, often including proofs, usually including nice examples and diagrams, and always including references to other sources. It's far from the bare minimum... $\endgroup$ Commented Apr 30, 2011 at 18:20
  • $\begingroup$ ...If you strung all the wikipedia articles relating to calculus together, what you would get would, I think, be comparable in length and superior in content to many of the standard American calculus textbooks. The reason is that a lot of different mathematically competent people have worked on them, and in a very collaborative way. $\endgroup$ Commented Apr 30, 2011 at 18:22
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Perhaps The How and Why of One Variable Calculus by A. Sasane, published by Wiley (see link below)?

http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1119043387.html

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For probability you can use "Probability and Measure", by Patrick Billingsley. Another option is "A Course in Probability Theory" by Kai Lai Chung.

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