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I am a pre-engineering student currently taking a Single Variable Calculus course at a community college.

I recognize that my future success (or not so much) as an engineer will be based, in large part, on my capabilities with and understanding of Calculus. Therefore, I really, really want to master it like I've never mastered any subject before.

I'm doing well in my class, and my instructor is great, but I am under the impression that this course and it's textbook (Calculus, Early Transcendentals by Stewart) do not delve quite as deeply into Calculus as I would like. Also, the textbook frequently introduces new techniques and concepts with little to no explanation.

(Incidentally, I'm a self-taught software developer, so I am adept at learning new topics on my own. Learning Mathematics is, IMHO, quite similar to learning a new programming language.)

So I'm hoping to find some really excellent Calculus textbooks that will give me deep insight into the topics of differentiation and integration (and any other topics my course may be missing).

I've used Google and my school's library to search extensively, and I've found no shortage of Calculus textbooks. My problem is that, since I'm just now learning the basics, I have no way to know just how in-depth an advanced or in-depth book should go, or what important information my current textbook may be missing.

I own a copy of The Calculus Lifesaver, by Adrian Banner, which is absolutely outstanding. If anyone reading this happens to be struggling with Calculus, this is the book to turn to.

I also have been taking advantage of the Calculus courses in MIT's OpenCourseware. Calculus Revisited, with Herbert Gross, has been very helpful. His way of explaining the concepts just really "clicks" with me.

So, with that said, I'm just hoping the experts in the community here can recommend some great resources (e.g. books, free online courses, or other media) to help me optimize my knowledge of Calculus. Thanks in advance!

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  • $\begingroup$ Maybe this answer of mine would help. $\endgroup$ – Venus Jun 7 '15 at 5:10
  • $\begingroup$ I would suggest "A Course of Pure Mathematics" by G. H. Hardy. It delves really deep into the subject matter and is specially designed for students of age around 15-16 years. Spivak's Calculus also does the same thing he needs too many pages and writes in a style which removes all the "inspirational stuff" from the subject matter. $\endgroup$ – Paramanand Singh Jun 7 '15 at 6:26
  • $\begingroup$ I agree with the answers suggesting Calculus by Tom Apostol, if you want a course that includes both theory (i.e. detailed proofs) and practice at a high level. If that's too difficult, Calculus by Marsden and Weinstein is a good alternative, as is A First Course in Calculus by Lang. These could be described as intermediate in difficulty and rigour between Apostol and Stewart. Lang has slightly more rigorous theory, Marsden and Weinstein a greater number of computational practice exercises, which nonetheless tend to be more instructive and varied than the ones in Stewart. $\endgroup$ – Keith Jun 7 '15 at 23:08
  • $\begingroup$ Let me add that Apostol has a number of real-world applications throughout the book, especially from physics; that's one of its characteristics. $\endgroup$ – Keith Jun 7 '15 at 23:09
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    $\begingroup$ @Venus Yes, thank you - that was helpful. My class begins integration in two weeks, and I've been reading ahead a bit, but I could definitely stand some more enlightenment in that area. $\endgroup$ – tommytwoeyes Jun 9 '15 at 3:32
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I recommend Advanced Calculus by G. B. Folland. It is rigorous and still elegant, especially when it comes to vector calculus.

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  • $\begingroup$ While all of the recommendations everyone has made are excellent, especially Spivak's Calculus (I've seen that book recommended frequently), I must chose this response, because one introduction to Folland's book (linked for posterity) that I read said it has an abundance of physics examples, which will be helpful to me as an engineering student. Thank you, everyone! $\endgroup$ – tommytwoeyes Jun 7 '15 at 20:15
  • $\begingroup$ Sorry to say, but in your link, it only says that Introduction to Calculus and Analysis by Richard Courant and Fritz John provides a lot of motivation with abundance of physics applications. $\endgroup$ – Henry Jun 8 '15 at 0:43
  • $\begingroup$ You may be right that I misunderstood the description, but I don't believe so. It said (this is copied & pasted): "The abundance of physics applications, make it ideal for physics majors and engineers..." I understood an "abundance of physics applications" to mean examples of physics applications. $\endgroup$ – tommytwoeyes Jun 8 '15 at 1:11
  • $\begingroup$ @Masacroso Have you checked this(amazon.com/Advanced-Calculus-Gerald-B-Folland/dp/0130652652)? $\endgroup$ – Henry Jun 21 '17 at 11:46
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From what I gather the holy grail of calculus books seems to be Calculus by Micheal Spivak.

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    $\begingroup$ Just keep in mind this book is for mathematicians, so you'd like to have another book where you can practice the methods described in here. $\endgroup$ – hjhjhj57 Jun 7 '15 at 1:20
  • $\begingroup$ That sounds right - some of the older Calculus textbooks I've looked at are so rigorous that I realized I would need a textbook on Mathematic Notation just to keep up. But, it wouldn't be a bad idea to read one of those, anyway. $\endgroup$ – tommytwoeyes Jun 7 '15 at 18:03
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If you want CALCULUS to be fun, you can also read these two along with Spivak

Don't just go on there names, these are not only for Dummies. Here is what author has to say- enter image description here

I personally like those books which talks to you while you are reading them. It does and it is humorous too!

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  • $\begingroup$ Since you like books with a more personal, colloquial tone than the typical textbook, you may like The Calculus Lifesaver book I mentioned in my question, by Adrian Banner. It is written on a personal level and contains frequent jokes, many of which are even funny. ;) $\endgroup$ – tommytwoeyes Jun 7 '15 at 18:06
  • $\begingroup$ Yup. I already ordered it. Thanks $\endgroup$ – Bhaskar Vashishth Jun 8 '15 at 3:44
  • $\begingroup$ Thanks for your suggestion. I've collected a couple "for dummies" books over the years, and they're usually pretty good, although the series title is the worst marketing ploy I've ever encountered. $\endgroup$ – tommytwoeyes Jun 9 '15 at 3:27
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I recommend the two volumes of Calculus by T. Apostol, they are a classic.

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  • $\begingroup$ Thanks - I will have to get that Apostol's books. So many people recommend it; not only here but all over the internet, so I reckon it must be really good. $\endgroup$ – tommytwoeyes Jun 9 '15 at 3:28
  • $\begingroup$ You are welcome, I am sure you will enjoy it, and I think it suits your purpose very well. I must confess myself, that it was only after many years that I came to realise what a jewel those books are. $\endgroup$ – Rogelio Molina Jun 9 '15 at 3:31
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When I was a freshman in college, we used Calculus by Tom Apostol. It took me many years to appreciate its greatness. However, if you'd like to learn the subject in depth, this book is really worth sinking your teeth into.

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  • $\begingroup$ I do - Calculus seemed intimidating at first, but only because of hype, I think. The more I get to know it, the more in awe I am. Thanks for the suggestion. $\endgroup$ – tommytwoeyes Jun 9 '15 at 3:29

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