# Spivak's Calculus?

I have seen many users here asking questions about problems in what they call "Spivak's Calculus Book". I have never seen the book, and information online is scarce. From what I've gathered, it is just a more rigorous calculus 1-3 book with harder problems. I have already taken calculus, and I am about to take analysis. Is there any point for me to buy Spivak's book, or is reading analysis books better at this point? Is this calculus book better than a standard analysis book? Why don't students who are gifted enough just start off with an analysis book instead of Spivak's calculus book?

• I think Spivak's book is supposed to prepare the ground for harder proof-based books. If you feel you have a solid background in calculus, I don't see any point in going for it - start analysis with an analysis book! Later on, please do check out Spivak's Calculus on Manifolds - it's a must-have, IMHO. – DrHAL Jun 3 '16 at 17:06
• The word "analysis" is a overloaded. There is functional analysis, complex analysis, real analysis. Spivak's book is none of that, but rather a book on calculus: limits, continuity, the classical "mean" theorems, Bolzano, integration and differentiation in $\mathbb R$, sequences and series. One might argue that the word "calculus" is overloaded, too, however. =) – Pedro Tamaroff Jun 3 '16 at 17:06
• Other books that do calculus consider things that might be more in the lines of introductory multivariate calculus (like Stewart's). To compare "calculus" and "analysis", consider looking at Apostol's Calculus and his Mathematical Analysis. – Pedro Tamaroff Jun 3 '16 at 17:07
• @DrHAL Well I'm sure there would be plenty of problems in Spivak's book that I wouldn't be able to solve, but I know the basics (what you learn in calculus courses). I just don't see the point of having a very rigorous calculus book. If a student is ready for a very rigorous calculus book, why not just go to an analysis which is presumably only slightly higher? – Ovi Jun 3 '16 at 17:08
• @Ovi One reason is the exposition. I don't think one can find a better mathematics textbook writer than Spivak. Furthermore, if one reads an analysis book, it is generally taken for granted that you know the mechanics of differentiation and integration. This isn't the case with Spivak, who presumes nothing from the reader. – MathematicsStudent1122 Jun 3 '16 at 17:10

## 2 Answers

It appears that there is tradition of studying calculus twice: first the non-rigorous version and then the rigorous version. Luckily I happened to study both at the same time (with school textbook being non-rigorous and Hardy's Pure Mathematics at home) and I did not find any benefit at all of the non-rigorous version.

I think the non-rigorous version is available because calculus is used in many applications and there is no need of a rigorous approach to learn the applications of calculus. Perhaps those who don't have to make a career in mathematics (or to use Hardy's phrase "students whose interests are not primarily mathematical") don't really need to study the rigorous version unless they would want to do it just for the sheer pleasure rigor adds.

But on the other hand if you have interests which are primarily mathematical it is a must to study calculus with rigor and the sooner you do it the better. I have myself studied Spivak's book but at a much later stage and apart from few nice problems (BTW all these nice problems of Spivak's book are available with solutions on MSE) I did not find much which was new compared to Hardy's book. But if you have never studied any books which treat calculus rigorously then Spivak's book is a very good option.

Your question "why not start off with an analysis book instead of Spivak's calculus book" appears to make sense on the assumption that "calculus" and "analysis" are on similar level the difference being lack or presence of rigor. This assumption is not entirely true. In most cases people are introduced to calculus at young age (high school age 16 years or similar) and the subject is treated in a concrete manner whereas most of the analysis courses (there are many real analysis, complex analysis, functional analysis and perhaps many more which I am not aware of) focus more on abstractions and some complicated and difficult topics like Lebesgue Integration, Fourier Series which are not covered in calculus courses.

In my opinion it is better to learn the usual calculus stuff (limits, continuity, derivative, integral) with full rigor and study advanced topics in analysis courses. Unfortunately many real-analysis books do cover these same calculus topics with rigorous proofs and then jump on to advanced / abstract topics. I personally don't like this redundancy and even the attitude that you can sacrifice rigor while studying/teaching mathematics.

• As much as I would like to agree with your perspective, in the U.S.A. at least, only a very small minority (probably less than 5%) of students who study calculus will eventually take a proofs-based real analysis course. In short, calculus books (in the U.S.A. at least) are not written for "mathematics students". – Dave L. Renfro Jun 6 '16 at 14:31
• @DaveL.Renfro: Story is same in India also. But isn't it strange that calculus is the only part of mathematics where there is something called a non-rigorous version. I don't see so much lack of rigor in other courses of high school like geometry trigonometry and algebra. Perhaps there is step fatherly approach towards calculus and this kind of describes my overall feeling about calculus education in general. – Paramanand Singh Jun 6 '16 at 15:22
• I think one of the problems calculus has that geometry, trigonometry, and algebra do not have is the enormous amount of material that is dumped into calculus. For example, analytic geometry (conics, rotation of coordinate axes, polar coordinates, 3-dimensional analytic geometry of lines and planes and quadratic surfaces, vectors, etc.), lots of work with limits even before the derivative shows up, numerical methods (Newton's law, trapezoid rule, Euler's method for simple ODEs, etc.), epsilon-delta (intro.), vector calculus, series, applications (work, pressure, volume, center of mass, etc.). – Dave L. Renfro Jun 6 '16 at 15:40
• @DaveL.Renfro: Fully agree and I like the phrase "dumped into calculus". Don't know why is syllabus of calculus designed that way? I am not into teaching so difficult to see what is the pedagogic advantage of all this "dumping into calculus". – Paramanand Singh Jun 6 '16 at 15:44

I personally think that Spivak is a perfect book to read just after calculus and to get ready for real analysis. Spivak's book will teach you the basic tools of analysis, and previous knowledge of calculus will help greatly to follow the book.

And, as mentioned in the comments above, the book is wonderfully written and a pleasure to follow. It has interesting topics and challenging exercises. If you have the time, I think that going through this book will be a good investment.