What are the best calculus books?

  • The problem with books like Thomas’ Calculus or Stewart Calculus is that you won’t get a thorough understanding of the inner mechanics of calculus. As long as you don’t have a good prof or teacher, I would stay away from these books. If you want to understand what I mean, take a look at some arbitrary sections in these books. You’ll see a short paragraph, which serves as an intro, then some boxes with formulas, then a few workout examples and then a bunch of exercises. This means, you will only learn HOW to you the formulas instead of understanding the WHY!

    My advice is, visit YouTube, search for Michael Van Biezen, learn the techniques of Calculus 1–3 (ca. 17 hours), and then, to understand the inner mechanics of Calculus, read Tom Apostol. Biezen will serve as a shortcut for learning the techniques and Apostol will teach you the WHY.

    Alternatively you can search for Prof.Leonard on YouTube and watch his Calculus 1–3 lectures (ca 168 hours). He works through the books like Stewart Calculus but tries to teach you the sections in detail. Nevertheless, I would prefer the first way Biezen -> Apostol.

    To answer your question,

    • Gilbert Strang – Calculus (very good, but in my opinion to conversational. You can find it for free on the website of MIT)
    • Tom Apostol – Calculus (very very good, but you need to put in serious effort)
    • Michael Spivak (didn’t read it, but many people say, it’s quite harder than Apostol, but still one of the best books to learn Calculus, although only Single Variable Calculus)
    • Serge Lang – First Course in Calculus (makes fun to read it, built more on intuition)
    • Thomas’ Calculus (short on explanations and too dry)
    • Stewart Calculus (same as Thomas’ with the exception, that he has more real world examples)

    If you don’t want to buy a hardcopy you can get a comprehensive Calculus book from OpenStax where Gilbert Strang is one of the Authors. (see link below).

    I hope I could help you. I struggled a lot with the same question.

    OpenStax

    If you are serious about calculus stay away from trash like Calculus for the practical man, Calculus made easy, anything by Schaum’s or anything flashy with lots of colors, pictures, jokes and all that useless fluff. All that stuff that pretends that calculus is a walk in the park or can be learned in one week is rubbish. Any book that spends more time on technique and applications, instead of core ideas and theorem-proving, are not worth your time and money. The worst of all is that dude S.Thompson (Calculus made easy) who even puts down serious authors for trying to present calculus in a formal and precise way.

    I’d go for serious authors who don’t take their readers for idiots, like Spivak, Apostol or Courant. There are other serious courses in calculus, but those are very formal, high-quality and popular.

    There are 2 authors that I consider amazing, and their calculus books

    1) Calculus by Tom M Apostol



    2) Differential and Integral Calculus by Richard Courant

    To build your high school concepts from scratch you can study calculus from Thomas,Calculus ,Pearson. I found this book very understanding and taught me a great deal. The examples were clear and built you up to the tougher exercises.

    I personally find the book T M Apostol, Calculus, Volume I and Volume II quite impressive. This book focuses on building the concept in a detailed manner. These books provide a solid, thorough treatment of single and multi-variable calculus. They cover all the important subjects of calculus like differentiation, integration, many different methods of integration. It can definitely feel dry at times but it’s not meant to entertain you anyway.

    You can further furnish your calculus by trying Rudin, Principles of Mathematical Analysis. It requires good understanding of basic concepts and mind-drilling.

    Good luck!

    You can also refer my answers : Sachin Sharma’s answer to What is the best way to learn mathematics and be excited about it as a high school student?

    Sachin Sharma’s answer to What are the best mathematic books for self-study and master the subject?

    There is an amazing book called, Thomas’ Calculus.

    It starts with the very very elementary stuff like what is a number etc. (obviously you can skip these chapters) and goes on to solving second order differential equations. It teaches you limits, derivatives, integrals. Everything you need to know! I can assure you, it’s wonderful. You will not want to look at any other book after starting it.

    It assumes that you know NOTHING, absolutely nothing at all about any of it, and teaches you from the scratchiest scratch you can think of. That is, from the very basics.

    I really like its explanations. They are very simple and obvious and very easy to grasp. There are plenty of colourful figures to supplement the text, in the margins.

    You can get the ebook from google. Just type “thomas’ calculus” and check the links there.

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    The best book unfortunately not well known I ever perused but never used as an undergraduate freshman in 1967–1968 was Calculus of One Variable by Joseph W. Kitchen(1968)! The probably unique “One Variable” in the title among the hundreds for a century would suggest that it stands out. This is a high quality Calculus text! The definitions and material were more in keeping with modern formulations of the last 30 years and presented at a graduate school level of that era but still very readable for a first time student well grounded in pre-Calculus and elementary high school math courses certainly moreso for an upper echelon math major. It was way ahead of its time! You could say it was a bit advanced nonetheless it was and is excellent for a very bright student!

    A little above “middle” level book, which we didn’t use in my undergraduate freshman days, for a first time non-high school Calculus student is Calculus With Analytic Geometry by Richard E. Johnson and F. L. Kiokemeister also from the 1960s but much much much less sophisticated than Kitchen’s, not in the best category, but can be described as very good and above average from the “stone ages”. It is the only beginner’s Calculus book or any book I have ever seen that contains the glossed over but pivotal, fundamental, critical, basic, essential, implied but never stated:

    Let f:W—>R, g:W–>R, R the reals, f(x)=g(x) for all x in W(the reals), except for x=k in W; the Limit f(x) as x→k exist => Limit f(x) as x→k=Limit g(x) as x→k. (This is my “isomorphic” version not what appears in their book). Seems trivial but it is absolutely indispensable for if we didn’t have it there would be no Calculus for most mappings need to be reduced or augmented to a better form to find its limit!

    The most famous book of the mid 1960s was Calculus with Analytic Geometry A First Course by Murray H. Protter and Charles Bradfield Morrey, Jr. both from U. C. Berkeley. It was the second best selling text in the U.S.! We didn’t use it in my Calculus classes. However Professor D. M. Bloom(Ph.D. Harvard in abstract algebra, but didn’t teach there) from my university used it. He was reported to be a perfect teacher in all undergraduate math subjects(I never took any courses with him). The fact that it was so well regarded including by Professor Bloom would indicate that this book was top notch.

    If we’re to judge by what arguably one of the five best mathematicians of the first third of the 20th century Godfrey Harold Hardy(of England) said that he fully appreciated analysis when he read and self-studied probably in his twenties the book by Camille Jordan: Cours d’analyse de l’Ecole Polytechnique(1882–1887). It is in French and unfortunately no English translation to date. If someone of Hardy’s caliber praises this it must be supreme! However this is unquestionably only for superior math students who can read French.

    A Course of Pure Mathematics(1908) by G. H. Hardy himself is an outstanding classic probably second after Jordan’s work of about 135 years ago.

    A superb text from the mid-1990s by James Stewart of McMaster University and University of Toronto is Calculus: Early Transcendentals. It is a large book about a foot long in length with over 1000 pages. This is in the same league as Kitchen’s with many wonderful applied real world examples from many fields not just physics; it’s at a higher level than Protter and Morrey’s book. Written no doubt for entering freshmen without spoon feeding or coddling where all prerequisites for this first course in Calculus must be met in high school and no remedial classes available at high caliber places of learning and therefore others attending a different near or same level quality higher institution should be readied in a parallel way! It is so good that basically you don’t need your professor or his graduate school teaching assistant! You can do it on your own!

    Finally I want to mention Khan’s Academy the videos and books if he has any: worthless at all levels from Calculus to his very elementary algebra high school material, etc.(If the purported strides were really there we’d see tens of millions of accomplished students which we haven’t and certainly his program would have been implemented immediately in the U.S. high school and college system as well as in Europe, Asia and elsewhere!; similarly the “New Math” failure of my generation about 1/2 century ago or the nonsense “hooked on phonics” 25 years ago to improve the deplorable state of reading!). Khan himself I’m sure is an excellent Ph.D. mathematician from Stanford but I see nothing of teaching value from his presentation. Bill Gates I think touted it for his kids when they were younger. He’s not one to judge a lowly Harvard dropout specializing in detestable boring programming and computer software(inspite of his then perfect 800 math SAT score!)

    It goes without saying that all of the above are not pre-Calculus works. They are for students at better 4 year universities.

    The best Calculus books are definitely needed and are an advantage but they will not nor have they created a sudden upsurge of students understanding this admittedly difficult and esoteric discipline! It definitely requires an above average intelligence and alot of persistence!

    If you are thinking of single-variable (i.e., introductory) calculus, I really like the book from which I first learned calculus (as a senior in high school). This is the one by Crowell & Slesnick, which I think may already have been out of print by the time I took it during the 1993–1994 academic year. I just googled it, and aside from being able to buy used copies, there was apparently a project to typeset it with latex about a decade ago. You can find the fruits of that labor at this website (though some of the references are broken, so that could cause some easter-egg hunts). The Crowell & Slesnick book is more rigorous than the cookie-cutter books, but it does this in an effective way, and in my opinion it is considerably better than the standard books that are normally in use. (Also, this wasn’t explicitly part of the question, but the cookie-cutter books for the introductory courses—for which changing the color of the cover and changing problems in a rather cosmetic way forces people to buy new editions—create a really obnoxious, unfair situation for undergraduates, who often end up needing to overpay for the current version of these books, which mostly haven’t changed seriously in a long time and which aren’t the best books on the subject anyway.)

    I first learned multivariable calculus from one of the cookie-cutter books. I think it was Edwards & Penney, but many others are really similar to that. I thought it was reasonable, but I can’t particularly compare it to others in the cookie-cutter crowd. I have taught out of some other one at some point, but I don’t remember which one. I went to Caltech for my undergraduate education, and the calculus books used there for time immemorial were “Tommy I” and “Tommy II” (aka: volumes I and II of Apostol). (However, when I TAed Math 1 during my junior year, other choices were used instead, including lecture notes by Barry Simon for Math 1a.) I skipped out of Math 1, so I only used Tommy II as a student. I was fine with it, though it was challenging, and I needed to get used to the level of rigor in the book. Less mathematically-inclined students, even at Caltech, often had trouble with Tommy, so it’s not for the faint of heart. (That said, I think it was a good experience for even the people who struggled with the book, because in education it can also sometimes be true that what doesn’t kill you makes you stronger. In my opinion, this is one of those situations.)

    There are certainly many books online, such as through MIT’s Online Textbooks, so take a look at what they have there.

    Thanks for the A2A.

    Let me relate my personal experience in mastering calculus. In the dark past, students were just beginning to study calculus in secondary school in the United states, so my first encounter with calculus was as a first year college student. It was a very difficult and thorough course. (My final grade was C-, which might bring some comfort to those of you who struggle with mathematics.) At the end of the course, despite my grade, I felt that I had a pretty good grasp of calculus.

    Then in my fourth year — this is the U.S. remember, I took a course called “Real Analysis”. The text was Rudin. At the end of that course, I said to myself, ”I was wrong earlier; now I understand calculus.”

    Then I went to graduate school. In my first year, I took a more sophisticated Real Analysis course. My undergraduate course had prepared me well for this one. At the end of that course, I said to myself, ”I was wrong earlier; now I understand calculus.” Then I taught calculus.

    For almost 40 years of teaching, I taught calculus nearly every term. I had brilliant student and mediocre ones. I answered routine questions and insightful ones. I developed a large store of explanation and examples I got so I could teach calculus over the telephone. Even now, retired about 16 years, if you asked me to give a lecture on some aspect of calculus, I could do it off the top of my head, complete with examples.

    So if you want to become “a master in calculus”, TEACH It.

    Oh my.

    They are very, very different.

    I own Spivak’s Calculus. Unless you are a supergenius, I would strongly, strongly advise you look elsewhere for an introduction. When I looked through it, I commented, “This is not a calculus textbook. This is a Real Analysis textbook with delusions of simplicity.”

    I actually recommend Spivak’s Calculus, don’t get me wrong, but not for learning calculus! It is an interesting approach to learning Real Analysis, and could even possibly be used to teach calculus by a highly gifted teacher who knows well how to translate.

    I don’t actually know Strang’s Calculus but I assume it is a far more normal textbook akin to Simmons or Stewart and thus far, far more suitable for a beginner.

    Good luck.

    I recall a book called “Calculus made Easy”. It was excellent, in that it did what it claimed, namely made calculus easy to understand, adopted an intuitive approach to understanding it and also did not compromise on the more rigorous aspects. A copy of it is available on Gutenberg Project.

    Sorry for not being able to remember the author. The book was pretty old.

    I also highly recommend KhanAcademy’s Calculus playlist, one of the best topics to learn from there.

    EDIT

    Thanks to Ananya Aparajita , the author of the aforementioned book is Silvanus Thompson.

    I personally found Thomas Calculus pretty helpful. It starts from some preliminaries, functions, all the way to a little advance parts of vector calculus, which is very helpful in physical applications.

    May the force be with you! 🙂

    My days of teaching calculus (and learning it) are long gone, but I am prejudiced in favor of more cookbook type approaches like Thomas for beginners rather than more mathematically minded ones like Spivak and Apostol. The latter are probably great for gifted students, especially if they have done AP Calculus in HS.

    Real mathematicians love Spivak, Apostol, and Courant, and they are great books for future mathematicians, but I’m not sure if they are for all. There is certainly a lot to like about Courant and John.

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    As an engineer, I would suggest the books which are less rigorous and more practical oriented if you are not maths major. During my college days, apart from usual advanced engg. maths text(Kreyszig) I had an exposure to Thomas’s Calculus, which is pretty decent. I have also gone through Calculus by Gilbert Strang, which I found one of the best till date. I did give a shot on Keisler’s infintesimal Calculus, but I got bored in 1st chapter, where he deals with hyperreal number in axiomatic way. In future, I am planning to read Stewart’s Calculus and Thompson’s Calculus (heard it was adopted by Feynman ). I fear I might never resort to Apostol…the Bible.

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