Looking for a book on Calculus

I’m looking for a book that emphasizes the theoretical aspects of Calculus. Especially one dimensional.
Right now I’m taking Calculus III at my high school. Last year I took Differential and Integral Calculus and received a 5 on the AB AP exam. I’ll be taking the BC exam this May, though I’ve already covered all the material on the exam.

I think I’ve had a great background in Calculus, but now that I look at colleges, it seems that the AP class puts less emphasis on the “theoretical” and more on the “computational”. We prove theorems all the time, but probably not as rigorously as possible. My #1 choice college strongly encourages all Math majors to re-take differential and integral Calc in their Honors course, which is much more theoretical. More individually writing proofs and such.

While I think that makes sense, the one thing I don’t want is to re-take single variable Calculus again. I’m covering a good portion of Multivariable Calculus this year, so I’d in essence be going backwards. Instead, I hope to place into their Analysis course. Of course, I don’t want to be unprepared, so I figure I could easily teach myself “theoretical” single-variable Calc on my own time. I have plenty of time before I start college.

If anyone has a suggestion for a book on theoretical Calculus of a single variable I’d appreciate it.

It may involve more then one varibale, but if your looking for a book more on theory, look for books on Real Analysis, IIRC, that class was more of a theory based class.

Here at the University of Chicago, there’s a first-year honours calculus course which (from your description) matches your desires for rigour, etc. The text usually used is Calculus, by Michael Spivak. Disclaimer: I’ve not actually ever seen this book; I’m just going on second-hand recommendations here. (In fact, I’m a physicist.) Perhaps one of our resident mathematicians will chime in and let me know whether this is a good recommendation or not.

You can also look at the webpage for the aformentioned course, which has a few other books mentioned.


I’d ask there instead.

You could always start out in the Honors course, and if you find it too easy, drop it and pick up something else. Talk to the professor, or the professor teaching a Real Analysis course. College course schedules are a lot more flexible than high school schedules.

FWIW, I was a math major, and I APed out of Calc 1 & 2. I took Calc 3 my second semester, but I didn’t get to Real Analysis until my Junior year.

Mike’s right: Calculus by Michael Spivak sounds like what you’re looking for. A great book, too.

My own biases are going to come through here very hard:

Screw single-variable analysis.

If you’re still reading, go pick up Spivak’s Calculus on Manifolds, which is a very thin book with a lot inside it. Single-variable calculus only makes good sense in the context of multivariable calculus. In fact, it’s an arbitrary distinction, since the essence is <ramble snipped>

Nothing against CoM in principle, but it’d be very tough to read it without first studying single-variable analysis, IMHO. It spends, for example, a total of 5 lines on the definition of a limit,

This is true, I suppose. I forgot that these days we coddle the calculus classes and don’t teach them what a limit is.

On the other hand, I still don’t advise a single-variable book if someone’s been through the AB/BC sequence. If they have to learn what a limit is, better to learn what a limit really is rather than this awful hack of doing it in one variable and then defining higher-dimensional limits in terms of that.

If you ever take a real analysis class (which is what you’re looking for), you’ll probably use Principles of Mathematical Analysis by Walter Rudin. May as well pick it up now so you don’t have to spend as much on books that semester.

that only Harvard course 55 freshies use it.

For a free non-standard analysis approach, here’s Elementary Calculus: An Approach Using Infinitesimals.

For delta-epsilon rigour, there’s always Apostol.

Wow, I wonder why?! (My #1 choice school happens to be U of C! I get my letter from them on Wednesday. How ironic!) Do you happen to know any math majors who could give me the low-down on the Honors class?

Thanks guys!

By the way, when I said “one dimensional” I meant “single-variable”. One-dimensional Calculus would be pretty boring.

If Amazon is haveing one of their extra-discount-on-paired-books deals, grab Real and Complex Analysis as well. For just pure real analysis, I’d actually go for Royden’s text, though.

You’re in B’more and UMCP isn’t your first choice for a math department? bah!

Go Terps!

College Park is merely my safety!

Go Maroons!

(Only kidding. I wouldn’t mind going there at all.)

I can get Spivak’s book used for $45 on Amazon. That seems like a good deal. MikeS’s link has all the homework problems and all the exams for the Honors class I want to try and skip/test out of. I think I’ll use a combination of the two.

Sure, I might not even end up at U of C, but it still seems like a good idea to know that stuff.

Another nice treatment of advanced calculus topics can be found in Shilov’s Elementary Real and Complex Analysis. The Dover paperback edition (ISBN 0-486-68922-0) is available for less than $20.

A few people have told me that Rudin’s text on complex analysis is only useful if you already know complex analysis. I can’t vouch for that, but it’s what I’ve heard.

We may be talking about different books. Real and Complex Analysis talks about significantly more than Complex analysis, as evidenced by the title.

Anyhow, I didn’t find it that difficult to learn from.

There are some interesting replies here, but I’d like to answer the OP a different way. What he said

suggests to me that what he may be missing, that this college wants him to have, is not necessarily so much the theoretical details of Calculus specifically, as an exposure to how rigorous, theoretical math works. They want him to get some experience writing proofs, etc. For most people, this is not something one can easily pick up on one’s own, especially from a book that already assumes familiarity with this kind of approach rather than one that’s designed to teach it.

Some mathematical writers are notorious for presenting proofs that are polished, concise, logical, but which give no indication of the thought process or how one might have come up with the proof in the first place; it helps to have someone to walk you through the theory and explain where everything comes from. And then, you have to write some proofs yourself, and have someone check them over for holes in the logic, hidden assumptions, incorrect reasoning, and so on. You can’t get this just from reading a book.

Many colleges, I believe, have a course they expect all math majors to take to introduce them to things like proof techniques, mathematical rigor, and some of the basic concepts that are used in higher mathematics. At some places, there may a course specifically designed for just this; at other places, a beginning course in real analysis or abstract algebra or something may serve this purpose. At the OP’s prospective college, it may be that this is what their honors calculus class is. My recommendation (along the lines of carterba’s) is to talk to the Math Dept. there and see what they suggest.