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13
The most enlightening Calculus books
By Antonio Cangiano. Filed under Math Education, Suggested reading
In Walter Rudin’s autobiography The Way I Remember It
Widely used calculus books must be mediocre. — W. Rudin
The temptation to discard that statement as elitist may be strong, but it is worth noticing how there is so much truth to it. Education is embracing a dangerous downward spiral towards the oversimplification of mathematics in favor of letting every student pass, despite their actual understanding of the subject.
Dumbing down mathematics is a dangerous trend which affects students at all levels, from primary school where children are no longer taught how to perform division by following the standard algorithm to a complete emphasis on anti-racist mathematics and calculus courses where delta and epsilon are not mentioned while teaching limits, because they are considered to be “too complicated” or “too confusing” for most students. Of course, this is not the case everywhere, and bright students who study on their own will always exist (despite the fact that the social satire of Idiocracy
This problem exists because educational changes such as these reflect negatively on the competence level of the average student, meaning that a decent chunk of the next generation of mathematicians, engineers and scientists will be facing increasingly complex, unsolved problems and challenges with an inadequate amount of preparation. Most students have a tendency to adapt their study levels around the difficulty of the examinations they will be exposed to. In other words, no matter how easy a Calculus course is, there will always be a majority of C students. Making the courses dumber has only resulted in the creation of artificial A, B and C students who wouldn’t stand a chance against a C student of the 50’s, in terms of understanding and mastering the subject.
It is therefore very important to consider what are the great classical books available that can provide a solid basis for studying mathematics and that can guide, inspire and enlighten the student who wishes to learn mathematics the way it should be. To be factual there are still institutions which adopt valid textbooks and didn’t jeopardize their curricula too much, but that’s not the general rule unfortunately. What are then the best Calculus books? “Best” is very relative to the student, but in this context I will consider what is “best” for relatively bright students who have a genuine interest in calculus/mathematics. Books that I myself have found very appealing to me, and therefore my list is declaratively biased. All of the following books do a very good job of covering the material, explain the subject matter well, have mathematical rigor, proofs for the most important theorems, challenging exercises, and are able to really teach Calculus I at a sufficiently high level. Each of the books listed below is able to enlighten, guide and inspire the reader who is willing to put the time and effort into going through them:
- Calculus
by Michael Spivak: I can’t praise this book high enough, to me this is THE calculus book. It has a great selection of topics, careful and extremely rigorous proofs, and it goes well beyond the scope of calculus, so much so that a better title would be “Introduction to the beauty of Real Analysis”, because it really bridges between calculus and more advanced real analysis, showing how beautiful mathematics is. Spivak literally guides you in a enlightening experience, discovering Calculus, starting from the properties of the numbers and building on top of it. The examples are very meaningful, the explanations are clear, and the subject is so well presented and motived. It is, in my opinion, one of the most inspirational mathematics books I’ve ever read. The exercises proposed at the end of each chapter test different levels of understanding and don’t fail to challenge the reader on the subject. These are exercises which really make you feel like you are taking a second course on the topic. In fact, I’d say that the exercise sections of this book are especially valuable, and you’re highly encouraged to work through all of them. This book is great for anyone who is serious about mathematics and who wishes to have a very solid foundations upon which to face the challenges of more advanced courses down the road. Spivak’s book comes with a few selected solutions to odd numbered problems, but if you are self-studying or are disciplined enough, you may want to consider the associated answers book (which by the way is in print, despite what Amazon lists on their pages).
- Introduction to Calculus and Analysis, Volume 1
by Richard Courant and Fritz John: a classical, well thought introduction to Calculus and Analysis in one variable, where explanations are very clear and the material is covered in a way which provides a good deal of motivation. The approach is more practical and less abstract than other Calculus books, while conserving a perfect balance between mathematical rigor and intuition. This book comes with plenty of exercises that will reinforce the knowledge of the student. The abundance of physics applications, make it ideal for physics majors and engineers as well. This is the first volume of a trilogy republished by Springer, if you are serious about Calculus, you may want to also consider the other two more advanced volumes: Volume II/1
and Volume II/2
. An absolutely beautiful set of books.
- Calculus, Vol. 1: by Tom M. Apostol: a very comprehensive book, methodical in the theorem/proof approach, adopted by many high-end universities as a first year textbook for courses in calculus or calculus and theory. Its coverage of the subject is impressive and provides a good selection of standard exercises. It is an excellent reference and textbook, albeit you may find it a bit dry and less inspirational than others at first, but you will eventually develop an appreciation of its teaching method. You may want to note that this is the first volume, and that the second volume is also worth getting: Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications
.
- A Course of Pure Mathematics
by G.H. Hardy: this is the 10th edition of a book first published in 1908 by the great mathematician G.H. Hardy. It is a classic textbook that brought much needed rigor, and reformed the way math was taught in the UK in the twentieth century. This book is intentionally written to address the brightest students of the time, therefore it is a very inspirational and enthusiastic piece of work with plenty of elegant proofs and suggestions, and exercises that are definitely very challenging. For example, there are exercises coming from the Math Tripos examinations (at Cambridge) from early part of the last century, and they will definitely keep you busy for some time. Some notations are a bit outdated and it is mostly an introduction to real analysis that may be a bit too much as a first book. But this book is a masterpiece nevertheless, and it’s a classic that as a mathematician you will want to have in your library. I think it’s ideal as a reference and as a supplement to other textbooks (e.g., Spivak).
More than just calculus, these also serve as introductions to Analysis, and in general to mathematics at an undergraduate level. They are challenging, not for the faint heart, but ultimately a joy for math lovers.
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32 Responses to “The most enlightening Calculus books”
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i think you missed Mathematical Analysis I.
I was hoping to see Apostol and Courant in the list. They were great back in 1980 when I had three years of Math in a six-year Engineering program, and it’s nice to hear they’re still going strong.
What about “Calculus Made Easy”?
I’ve kept a barely used copy of Calculus, Steward 3rd edition over the years. However it wasn’t until recently when I read through Calculus for Dummies that I realized how poorly my teachers covered some of this material.
Michael Spivak’s book is a real gem. I still have my battered old copy from college. Just leafing through it now (yeah, I still have it in my office!), I can remember how much “sense” it made. He really takes you from first principles all the way up.
Of course, it’s not actually the right way to teach maths - I prefer the “debugging” approach of Imre Lakatos, but if you get a kick out of the intricate details (and I do), then Spivak is one of the best.
Howdy,
Book 4 is available for free. Go to
http://www.archive.org/details/coursepuremath00hardrich
and download it. Sure, a later edition is nice, but take a look at the free one first. I have never seen the later printings, so comments (or commnets for users of this website) as to the pertinent differences would be good to see.
I’m with pippen, “Calculus made Easy” is a great read, it’s very interesting, even if the approach is considered by some to be heretical…
Well actually, that’s a draw for me
“What one fool can do, another can.” –Ancient Simian proverb
[ Note: your point about education is diminished by the two obvious typos on the comment submission form: "commnets" and "subsribes". ]
I used the Apostol texts in the 1980’s and agree that they are dry but good. As an added bonus, both my Vol. I and Vol. II copies were autographed by Paul Erdos during a brief visit at my university!
I’m surprised that Rudin’s own Real and Complex Analysis isn’t on the list.
@noway, @pippen, @snifty:
I’ve only listed books that I’ve personally used, but thanks for your suggestions.
@FreeIsGood:
Thanks a lot for pointing out that there is a free version available. I knew that but I forgot to mention it.
@jbgreer:
The typos were part of a Wordpress Template that I’ve downloaded. I didn’t make those mistakes, but thank you for spotting them.
@tec:
IMHO, “baby-Rudin” and “big-Rudin” are fantastic books but way too advanced for an introductory calculus course.
I’ve got to vote for “Calculus Made Easy” too. It truly is the first thing to read — before even signing up for a Calculus course. Reading it before class would have saved me much trouble in college.
I don’t like Spivak’s Calculus book, it’s not really a bad book, but I think it’s overhyped. Personally I would suggest Introduction to analysis by Norman B. Haaser, Joseph P. LaSalle and Joseph A. Sullivan. However it’s not a very popular book and somewhat hard to get. It starts with almost no required knowledge of Calculus and goes through everything that you would expect from a Calculus book but with the formalism you would expect from an analysis book.
so, “hard” is somewhere correlated with “better”?
c’mon; that argument is as weak as can be.
is it better to attempt to teach only those that seem to have “special” ability with math?
kind of like a homeless shelter saying “we’ll only feed those that remember to bring a fork and spoon”, isn’t it?
I used to be terrible at calculus all the way through highschool because most of the time we were told to just apply memorized formulas to different types of problems. I barely passed because I had this mental block because I used to wonder why the heck I applied formula X instead of Y??
When I got to college and got a really good teacher, the whole thing totally clicked in and I totally aced two calculus courses. It is all in the way you teach it. Some maths teachers are just plain lazy and don’t bother what calculus is all about.. when you see what it can be used for and how it is useful and why things are done the way they are, it becomes insanely interesting..
p.s. thanks FeelsGood for posting the link to the free text.
Anyone who has ever read Rudin knows he was a poor bookwriter.
Spivak’s Calculus is hands-down the worst book I have ever read in my entire life. I still have my copy because I can’t figure out a good enough way to destroy it. For those who know a little math, I would add this: His treatment of manifolds, a wonderful, graphically intuitive topic, is 3 full pages of definition. I had never seen them before this. I never had such an easy subject made so difficult by such bad writing.
Hardy is almost as bad as Spivak! Some bastard gave me that book when I was 14 or 15, and it was supposed to be my self-taught introduction to number theory. I didn’t get the subject at all until a much better teacher with a much better book made it clear. After that, it became my specialty.
These are THE two books that most killed my enjoyment of mathematics as a career. If you want to stop being a mathematician and go into something else (such as CS), these are the perfect books for you.
At the risk of offending the purists, a fun introduction for the more visual students is Prof. E McSquared’s Calculus Primer - and other books in the series.
Addition to previous post:
It occurs to me that we are probably talking about different Spivak books. I was thinking of Spivak’s Calculus on Manifolds. Do not ever read that book.
Spivak’s Calculus, I don’t know. Based on style alone I would guess it’s awful, but I won’t pretend to know.
Calculus is a pillar of Western civilization.
I’ve studied calculus at several levels, taught calculus in college, applied calculus in industry and government, and published peer-reviewed original research in calculus.
The basics of calculus are really easy: The ‘derivative’ is just the rate of change or the slope of a tangent line, and the integral is the area under a curve. The fundamental theorem of calculus is that essentially differentiation and integration are inverses of each other, each undoes what the other does.
A big point is that both differentiation and integration are linear ‘operators’.
While the above is true, it is not sufficiently deep or full to permit much utility.
To learn calculus, it can help to have a good high school level understanding of trigonometry. Understanding conic sections — parabolas, ellipses, hyperbolas — can also help,
Since the original motivation and the main early applications are to motion of objects in physics, it is helpful also to study the physics.
There are shelves of okay introductory calculus books. I learned introductory calculus mostly from
Richard E. Johnson and Fred L. Kiokemeister.
and taught introductory calculus from
Protter and Morrey
Both are okay as first texts.
Looking for a first text that will do well for all purposes is hopeless. No first text can give very much; more is needed. In particular, it is important to cover calculus as theorems and proofs, but these are too difficult for a first book.
For a student, the first book that emphasizes proofs should likely be one on linear algebra. There are many good books, but the classic is Halmos, good for a second or third book on linear algebra.
For the calculus proofs, Rudin’s ‘Principles’ is rock solid. There are at least three editions, and the editions get less severe and easier to read as they go on.
With the proofs in hand, really should take a detour and study ordinary differential equations. My favorite is Coddington.
Also, ‘calculus’ concentrates on the Riemann integral, and it is now long since past time to take more seriously the Lebesgue integral. For this, good sources are the first few chapters of Rudin’s ‘Real and Complex Analysis’ or Royden’s ‘Real Analysis’.
The teaching of calculus of several variables, especially using only Riemann integration, remains a mess. Mostly the physical sciences want to stay with 19th century approaches; mostly colleges want to avoid the subject; mostly the texts with careful proofs want to do calculus on manifolds and exterior algebra.
Of these approaches, the most useful for most applications remains the 19th century approach, and an excellent presentation is in
Tom M. Apostol, ‘Mathematical Analysis: A Modern Approach to Advanced Calculus’, Addison-Wesley, Reading, Massachusetts, 1957.
It’s charming and of crucial importance for a lot of physical science but omitted from the next edition.
With such a background, important applications include classical mechanics in physics, classical electro-magnetism (Maxwell’s equations) in physics, linear filtering in electrical engineering, optimization (Kuhn-Tucker conditions), and control theory.
Fourier theory and Hilbert space — e.g., as for quantum mechanics — are best done using Lebesgue integration.
Probability is often mentioned as an application of calculus, but Riemann integration is a clumsy approach. The good approach is as in A. Kolmogorov based on abstract integration based on the Lebesgue integral on abstract measure spaces. Good texts include Breiman, Neveu, and Loeve.
I have a couple Calculus books around and my favorite read is certainly Spivak. For just looking up how to do something specific, other books can do better, but for the sheer joy of the math, Spivak is the winner. Sadly, my copy is almost falling apart after many years of use.
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How about some recommended books for Calculus II, III (multivariate), and number theory? Or even some recommended books for other topics in mathematics like discrete math, mechanical physics, electromagnetic physics, quantum physics, computer science, etc.
Great suggestions for Calc I, however!
Thanks Matt, I plan to write a few more articles about different and more advanced topics.
[...] The most enlightening Calculus books (tags: book math) [...]
Well, if you are onto mathematics and want to learn ‘calculus’ more deeply, then consider studying ‘analysis’, which is the name mathematicians give to anything that’s related with limits and convergence (in fact, analysis is calculus made harder
well, not harder but is a rigorously formalization of it).
Want some good books for analysis? well, I don’t remember but I think that Analysis I from Serge Lange, and Analisis from Elon Lages Lima (portuguese book) are good sources of information to learn this topic
Dear Hearts;
It’s truely sad what’s become of calculus courses im the US today.I’ll throw my 2 cents in: I think a calculus course should be equal parts theory and applications.Making a calculus course all theory a la Landau is really doing a disservice to students by denying the power of an enlightened approach to applications.There is no pure math,no applied math-there is merely mathematics and some of it happens to be applicable.That’s why although Spivak is beautiful,I feel it would need extensive supplementation to use in a calculus course.It’s simply to “pure” and austere-too much of the body of calculus’ beautiful applications to the physical sciences is missing.Hardy is an analysis text-and I think it’s too old fashioned to be easily read today.
Now: My favorite calculus books:
Calculus by Edwin E.Moise,second edition:This book is a forgotten marvel by one of the great teachers and mathematicans of the 20th century.It presents rigorously and with no watering down the complete theory of single variable and multivariable calculus with many applications to differential equations,physics and chemistry-as well as complete,beautiful chapters on complex numbers,analytic geometry,trigonometry and linear algebra needed for college calculus.It also contains further appendices that develop some subjects further,such as set theory and the idea of measure.A crystal clear and COMPLETE presentation that none of today’s books can match and would prepare students for any real analysis course that followed.That this book is out of print is a crime-Dover would do the academic community a godsend by reissuing it in a nice,cheap paperback.
Practical Analysis In One Variable by Donald Estep: This to me is the closest book there is to a perfect calculus text,despite having analysis in the title.The rigorous development of limits,functions and number systems is developed in the context of mathematical modeling and numerical analysis-thus removing the barrier between pure and applied mathematics and getting both math majors and physics,engineering and chemistry students to think about mathematics PROPERLY.The book has just one defect to me:Estep has a wierd attitude that infinite series is an outmoded topic in real analysis and matters only in complex analysis-so he drops it.This bizarre conclusion aside which is easily supplemented, this would be my book of choice for a strong calculus course supplemented by Moise.
These are my 2 biggest choices.More later on other texts.
Andrew L.
to mlylecarlin… then what is your preferred book?
And what would you advice for understanding and solvin differential equations?
If you want to know why you’re having (or had) problems with epsilon-delta, infinite series, or infinity in general you’d best read Lakoff and Nunez’s “Where Mathematics Comes From”.
Lakoff and Nunez show precisely why mathematical ideas are not intuitive yet how mathematical ideas _must_ be grounded in the real world. They kill off Platonic theories of mathematics once and for all, which is a good thing.
The book goes into considerable detail (excruciatingly sometimes). Reading it yields one “Aha!” moment after another.
http://www.amazon.com/Calculus-Made-Easy-Silvanus-Thompson/dp/0312185480
Calculus Made Easy, easily the most intuitive and readable calculus book ever.
You cannot learn a new language from a reference book of its grammar. Native speakers are little aware of grammar and yet totally fluent in their use of the language. Mathematicians formalise a new branch of mathematics after its creative explosion has died. They then teach us the formalised form. Any field of knowledge grows from 0 to maturity. This is the way knowledge come to the human mind. I wish mathematicians can teach us calculus (or anything else) by following this journey of discovery. Why did Newton needed calculus? What problems were facing him? All the way to the formalisation of calculus in Analysis. Most maths books are just records of the authors’ current knowledge and understanding rather than a help to anyone. They are just “reference grammars”, not much use to people who wish to speak the new language.