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I stumbled across the book when searching for rigorous alternatives to Rudin with some solutions. It’s an “old school” (1965) calculus text but, I think, covers similar material to Rudin in a more chatty way. It also has solutions/hints to selected problems, and fewer than Pugh. (I have a hard time knowing which ones to solve, since there are 80+ per chapter).

Does anyone have thoughts on where this book stands vis-à-vis Rudin, Apostol 1 and 2, Spivak, etc? There wasn’t much information on the Amazon review page

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    $\begingroup$ It's fairly similar to many (well over 15) books roughly titled "advanced calculus" designed for a 1-year course at the 3rd-4th undergraduate year, a course that used to be very standard at most every U.S. university from the early decades of the 20th century to the 1970s, after which these courses (which probably still exist) began being phased out. I know this doesn't tell you much specifically about Buck's book, but my point is that stumbling across a book like Buck's somewhat like stumbling across a tree while looking for big plants in a forest. $\endgroup$
    – Dave L Renfro
    Jun 11, 2022 at 20:14
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    $\begingroup$ For older books, you're better off looking at published reviews, since amazon reviews are rarely written for books that aren't relatively recent (and haven't been recently reprinted, for example by Dover Publications). For some published reviews, see these at JSTOR. There are almost certainly other published reviews (e.g. Jour. London Math. Society and Canadian Math. Bulletin used to publish book reviews), but at the present time there isn't an easy way to locate them. $\endgroup$
    – Dave L Renfro
    Jun 11, 2022 at 20:18
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    $\begingroup$ FYI, Loomis/Sternberg and Nickerson/Spencer/Steenrod are for what one might call "honors level advanced calculus", and are significantly above the level of the standard advanced calculus text (e.g. Taylor/Mann and Kaplan and many others). These books differ somewhat from those used for upper level real analysis courses by including a nontrivial coverage (continued) $\endgroup$
    – Dave L Renfro
    Jun 11, 2022 at 20:53
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    $\begingroup$ of multivariable calculus (Rudin also has this, but his book is primarily used for the earlier proof-focused 1-variable stuff), along with a few physics applications and vector analysis (and sometimes some complex variables). Those older advanced calculus courses tended also to be taken by a wider class of students (physics, engineering, statistics, etc.) than the typical 1-semester or 1-year real analysis course. Incidentally, the published reviews of Buck's book probably focus mostly on comparing the book to other similar advanced calculus books the reader is expected to be familiar with. $\endgroup$
    – Dave L Renfro
    Jun 11, 2022 at 21:02
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    $\begingroup$ A later edition of Buck's text is on the MAA's Basic Library List of books recommended for college libraries. maa.org/press/maa-reviews/advanced-calculus-4 I have fond memories of the book from my first undergraduate course in analysis (we didn't have a transition course, so this and abstract algebra were our first proof based courses.) We used baby Rudin as a secondary text for that course. $\endgroup$
    – Brian Borchers
    Jul 3, 2022 at 15:57

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