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Can you suggest me some REALLY hard books on calculus and analysis. By hard I don't mean difficult in explanations, but with extremely challenging exercises (all worked out if possible) and useful insights and tricks. Also, I would like you to share similarly defined books (or handouts) about other disciplines in an undergraduate course (that is to say, the subjects mentioned here).

Thank you.

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    $\begingroup$ Can you provide an example of something close to what you are looking for, either at around the right level of difficulty or around the right format, perhaps for a different subject? It would help to clarify the question by replacing the bolded "really" and "extremely" with "at the level of..." or "harder than...". $\endgroup$ – user173 Jul 10 '14 at 11:59
  • $\begingroup$ Let's see... lots of people consider Spivak's and Apostol's difficult. I'm looking for much more than that. $\endgroup$ – user10024 Jul 10 '14 at 13:02
  • $\begingroup$ Of possible interest might be this take-home test of mine that I gave several times in the late 1990s to a post-BC calculus high school class. At least a fourth of the students now have Ph.D.'s in a science or engineering field. In fact, I believe nearly 2/3 of the Fall 1998 class does, including one with a Ph.D. in math from Rice University. (See the 2nd paragraph of this post for more about that phenomenal class.) $\endgroup$ – Dave L Renfro Jul 10 '14 at 18:46
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    $\begingroup$ But is calculus supposed to be "hard"? I thought it's virtue was that it was "easy", and could do so many things... $\endgroup$ – paul garrett Jul 10 '14 at 21:52
  • $\begingroup$ Out of curiosity, when I got home yesterday I looked at my class rolls for the classes having that home test (or a similar one), and I should probably revise my earlier claim of "a fourth" to $15$% to $20$%, and $2/3$ should probably be changed to $1/2.$ On the other hand, if I include everyone with any (science or non-science) graduate or professional degree (Masters, MBA, law school), I would guess it's over $80$% of the students from all the classes that got a version of that take home exam. $\endgroup$ – Dave L Renfro Jul 11 '14 at 14:52
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My pick would be Volume 1 of Richard Courant and Fritz John's Introduction to Calculus and Analysis. For discussions of how Courant/John compares with Spivak, Apostol, and other books, see the math StackExchange question Difficulty level of Courant's book.

Volume 1 of Courant/John does not have any solutions to the exercises, but complete solutions to all the problems were separately published by A. A. Blank:

Albert Abraham Blank, Problems in Calculus and Analysis, John Wiley and Sons, 1966, x + 264 pages.

You might also want to look at the list of honors calculus books posted in the following math StackExchange question:

Joseph Kitchen's Calculus (reference)

(ADDED NEXT DAY) Regarding Joseph Malkevitch's answer, yesterday I was thinking of mentioning some books from roughly the 1880s to 1890s, when textbooks in algebra through calculus were, on average, pitched at the highest level, but I didn't get around to it. Throughout the 1800s there was a gradual overall trend towards more difficult texts until the end of the 1800s, at which time there were several "downward adjustments" (I'm mostly thinking of the U.S. and England) due to various reforms and the increasing percentages of students going to college. (This is discussed in Châteauneuf [1] for those wishing a reference.)

Off-hand, I can think of three authors from this time who wrote calculus treatises at a fairly high level (in the algebraic-manipulative-mechanical sense, not in the modern rigorous sense), and with the beauty of the internet now-a-days it only takes me a few minutes to track down freely available digital copies of their books. See [2] through [8] below. I think you'll find plenty of "extremely challenging exercises" in these books, and many will be on topics you are probably not familar with. The books by Edwards are probably the most extreme in this respect, and over the years I've read book reviews in old journals that pretty much say this (while also being very critical of Edwards' lack of rigor, especially in reviews written after the mid 1890s).

[1] Amy Olive Châteauneuf, Changes in the Content of Elementary Algebra Since the Beginning of the High School Movement as Revealed by the Textbooks of the Period, Ph.D. Dissertation (under John Harrison Minnick), University of Pennsylvania, 1929, x + 191 pages.

Also published by Westbrook Publishing Company in 1929, and reviewed by Lao Genevra Simons in Mathematics Teacher 24 #1 (January 1931), 58-59.

[2] Joseph Edwards, An Elementary Treatise on the Differential Calculus

[3] Joseph Edwards, A Treatise on the Integral Calculus, Volume 1

[4] Joseph Edwards, A Treatise on the Integral Calculus, Volume 2

[5] Isaac Todhunter, A Treatise on the Differential Calculus

[6] Isaac Todhunter, A Treatise on the Integral Calculus and Its Applications

[7] Benjamin Williamson, An Elementary Treatise on the Differential Calculus

[8] Benjamin Williamson, An Elementary Treatise on the Integral Calculus

Incidentally, solutions manuals to older texts (in English) were often called Keys, and I'm sure some of the texts above have keys (typically prepared by someone other than the text's author), which I'll leave you to search for if you're interested. I did happen to come upon the following Key while looking up the books above, so I'll include it:

Hunter, Key to Todhunter's differential calculus

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  • $\begingroup$ Dave I've only managed to download the first three. How would you download the others? They have "PDF (Google.com)" but that's not quite downloadable. $\endgroup$ – Mark Fantini Jul 12 '14 at 0:45
  • $\begingroup$ Wow, I just scanned through a couple dozen pages in Edward's on integration... I am impressed. That would be an evil source to mine for homework problems :) $\endgroup$ – James S. Cook Jul 12 '14 at 4:54
  • $\begingroup$ @James S. Cook: FYI, some other comments I've made about the Edward's book on integration are here and here. Also, I cited Edwards several times in my answer to the math StackExchange question Solving this integral?. $\endgroup$ – Dave L Renfro Jul 14 '14 at 13:53
  • $\begingroup$ @Fantini: It may be that you need to establish a google account to read the others. If this isn't possible for some reason, I suggest simply googling the author's last name along with the title I gave in quotes (i.e. a google-phrase search). I suspect this will lead you to other digital copies, and you might have better luck with one of these other digital copies. $\endgroup$ – Dave L Renfro Jul 14 '14 at 13:59
  • $\begingroup$ @DaveLRenfro I love that your answer is not the accepted answer to that MSE question. Ha ha. Nice work. $\endgroup$ – James S. Cook Jul 14 '14 at 17:04
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I learned calculus (by self-study) from the book by Granville, Longley, and Smith that Joseph Malkevitch mentioned in his answer. Later I had to work considerably harder on problems from Polya and Szegö's "Aufgaben und Lehrsätze aus der Analysis", which I believe now exists in English also.

Edit: The English translation is entitled Problems and Theorems in Analysis (links to one and two).

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For Calculus, I suggest: http://www.artofproblemsolving.com/Store/viewitem.php?item=calculus (which has an available solutions manual as well). Many of the problems in the book are collected contest problems. Also each chapter has a challenge problems section after the review problems.

Beyond that, although not limited to a particular discipline like a textbook, I recommend the Putnam Competition problems and solutions.

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  • $\begingroup$ Is there something prompting the downvotes on this response, other than just not agreeing with the suggestions in it? $\endgroup$ – PurpleVermont Jul 11 '14 at 0:06
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    $\begingroup$ Seemed like a perfectly fine answer to me. $\endgroup$ – James S. Cook Jul 11 '14 at 1:24
  • $\begingroup$ It's in a different vein than the other answers, but "contest math" goes a long way toward mitigating the compartmentalization otherwise often seen in math texts (as lamented here at the K-12 level matheducators.stackexchange.com/questions/3938/…) $\endgroup$ – PurpleVermont Jul 12 '14 at 1:06
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    $\begingroup$ I have more concern for an over-emphasis of problem solving. I see many courses offered with the attitude that proofs don't matter, or, at a minimum, the students know they can ignore the proofs because what is tested is the homework more or less. I often tell a story about how I turned the table on the process as an undergraduate. I challenged a prof that a proof would not be on the exam. He said, why? I said, because we're not tested on proofs. Well, it wasn't on that test, but it was on the final. Imagine the shock of my classmates. I digress... $\endgroup$ – James S. Cook Jul 12 '14 at 5:02
  • $\begingroup$ I don't recommend problem solving at the expense of proofs at all. But solving "non-routine" problems such as the ones typically found on something like the Putnam is a great way to see whether students really understand and can apply what they've learned, and not just parrot back problems "just like" the ones they've seen so far. And it can make them combine skills from different "chapters" or courses, which IMO is the whole point of learning those things in the first place. $\endgroup$ – PurpleVermont Jul 12 '14 at 17:45
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Richard Courant's Differential and Integral Calculus has both unusual "content" and exercises. Generally, speaking Calculus books from the turn of the century had mechanistic problems which were quite difficult. Many of these books survived into the 1960's including the Calculus book by Granville, Smith and Longley which appeared in 1904 with Granville as the only author.

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Some older diffyQ books may be of interest to you also (Boole, Taylor (Treatise, NOT Theory), Piaggio).

Note that older books were much more commonly containing answers. This is an area where modern practice has become LESS liberal, not more so.

You might look at Schaum's Outlines as they contain a lot of solved problems (sometimes answers, sometimes whole problem). I don't know they are "hard". But you can pick through them.

Thomas and Finney prior to early 80s had all the answers.

You may also consider to look on the web for old (prior to 1930) Cambridge Tripos exams.

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