Which Calculus Book? Stewart or Spivak?

[DickCarverPI]

Hey all, I'm going to be reading a calculus book after I finish my book on Trig. Problem is, i can't decide which book I want to read. I'm wondering if anyone has any insight on either book, or could recommend a better book. I want to use my time the most effectively so any input will be a big help.

Thanks in advance,

Dick

 

[snipez90]

Well, do you want to do calculus or real analysis? Spivak is for all intents and purposes an introductory real analysis text. If you think you might major in math or physics in college, you're probably best off just starting with Spivak.

 

[DickCarverPI]

Pardon my noobery, but what's the difference between calculus and real analysis?

 

[snipez90]

http://en.wikipedia.org/wiki/Mathematical_analysis

Roughly speaking, basic real analysis is the rigorous theory behind the concepts in calculus. In a typical first calculus course, the emphasis is on the mechanics. You can gain a lot of intuition for the concepts this way, but there are also several potential pitfalls when you don't have the added rigor of analysis. Honestly, going through either subject on your own for the first time is probably of similar initial difficulty, so if you really think you might wind up majoring in math or physics, do analysis now.

 

[Robert1986]

I'm sorry to say that I strongly disagree with snipez. When I took my first calculus course, I was really annoyed that the textbook (I used Stewart, btw) didn't go into (what I thought) was enough detail. There is a lot of "the proof of the next theorem is beyond the scope of this book" sort of stuff. As snipez said, calculus books are going to give to a lot of intuition, but not a lot of (good) theory. IMO, this is fine. I am currently taking Analysis I, and I realize that if I didn't have the intuition that I have from calculus, I would be totally lost. There is a reason that universities require that students take calculus before analysis. In theory, you could read most any analysis book, and understand the theory behind calculus. However, IMO, it is going to be needlessly difficult (and I see no real advantage) if you do the analysis first. Additionally, the Stewart book is going to teach a lot about different applications of calculus, it has really good problems, and really good mini-projects. There are two solution manuals for the book and both are very good.

 

[Nabeshin]

Personally, I think Stewart is complete rubbish. It exemplifies the type of "paper mill" textbook publishing which seems to be common these days. It seems he puts out a new edition every few years, and university students across the country are then forced to cough up the $100 for a new copy. The book seems very watered down, and upon reading through it I felt like the target audience was med school prospects, rather than physicists or mathematicians. (Now that's fine, if you're one of those then it's probably perfect!)

That said, it does seem to be the standard text for freshman calculus courses, so clearly somebody likes something about it. Comparatively, Spivak really does not seem like the type of book which should be used lightly, as it assumes a bit more mathematical maturity than most students seeing calculus for the first time have.

A final thought: I don't know what your purposes for learning calculus are, but the two books rather diverge there. Stewart is likely sufficient for a physical sciences undergraduate, and 3/4 physics majors. For those 1/4 of physics majors who are more interested in the mathematical theory, Spivak is a good bridge into upper division mathematics (real analysis, chiefly. But more than just that it serves as an introduction to rigor). For mathematics majors, Stewart is fairly useless since the majority of Stewart is examples in how to compute derivatives or integrals, and mathematicians do not really concern themselves with these trivialities.

 

[DickCarverPI]

Wow, thanks all y'all are really helpful. Nabeshin, that was exactly what I was looking for. I'm a Senior in university, non-math, non-science major and I'm just looking to educate myself in math. I want to understand the nuts and bolts of calculus, and then go beyond so I think I'm going to read Spivak next.

Thanks again,
Dick

 

[Robert1986]

To Nabeshin's point, I have to admit that I agree with your analysis of Stewart's book. But, I REALLY want to know where you were able to get a copy for just $100! Mine cost $250! The idea being that it can be used for Calc I, Calc II, and Calc III. But, I took Calc I in the spring, and by the time I took Calc III, the math department switched to another book. So, I have two $250 calculus books. I like learning the different integration and differentiation techniques/applications. Sure, this isn't exactly "pure math" but it is exactly what should be taught, IMO in a calculus course. I attend a major Engineering University, and I think you can understand why we use Stewart's book. I just can't see using Spivak as an introduction to Calculus, though. Instead of going the Spivak route, try Courant's two volume work. When I was in Calc. III, I started to plough through this. It is very interesting and presents the material in a "backward" fashion with respect to most texts.

 

[lurflurf]

+1 Stewart=rubish
Book titles are not a reliable indicator of content. Analysis and rigorous calculus are not interchangeable. Spivak is a calculus book suitable for readers with no knowlage of calculus. One does not need to learn wrong calculus ala Stewart to learn correct calculus later, in fact it is better not to. It is fine to do some nonrigorous calculus first, it is not needed for all, but might be helpful to some. If is also reasonable to read Spivak together with another book. Elementary Real and Complex Analysis by Georgi E. Shilov is a nice book at approximately Spivak level as is Introduction to Calculus and Analysis by Richard Courant and Fritz John.

 

[lurflurf]

Mine cost $250!
I like learning the different integration and differentiation techniques/applications.
I just can't see using Spivak as an introduction to Calculus, though.

Stewart is not worth $2.50.
You don't have to choose. Theory, methods, and applications mix together well, it is just a matter of finding the right balance. Spivak is not all things to all people, it has a very limited and specific purpose and audience.

 

[Robert1986]

I guess I just have a warm spot in my heart for Stewart :).

Anyway, I don't really know why you consider Stewart to be "wrong calculus". Can you explain?

 

[snipez90]

I just can't see using Spivak as an introduction to Calculus, though. Instead of going the Spivak route, try Courant's two volume work. When I was in Calc. III, I started to plough through this. It is very interesting and presents the material in a "backward" fashion with respect to most texts.

I think the difference between Courant and Spivak is simply that Courant includes applications (physics) in addition to the theory. I have read a decent amount of Courant volume I and his analysis proofs seem terser than those of Spivak. Spivak gradually guides you through the epsilons and the deltas, always starting with the intuition from a first course in calculus. For instance, he doesn't start the chapter on limits of functions with the epsilon delta definition, but rather an intuitive definition that anyone who had never seen limits before could grasp. Then he slowly begins formalizing each part of the intuitive definition after providing several geometric illustrations and explicit numerical calculations.

If you mean that the first few chapters of Spivak might put beginners off, that certainly may be true. Few people are thrilled about learning the field axioms, but you have to get the basics down if you want to get anywhere (learning the alphabet wasn't that interesting, but it opened several doors; yes I stole this from Herstein). Well if you are very comfortable with algebra and inequalities (the field axioms shouldn't be new, but there are subtle points), you could just skip to the interesting stuff then come back later. However, I think it's best to deal with potential misconceptions of the very basics right away before you get to the actual calculus.

 

[lurflurf]

The most wrong thing about Stewart is calculus is presented as a bunch of rules to swallow and regurgitate rather than something to think about and understand. It is what students expect these days. This is not unique to Stewart which is above average among bad books, but why bother. Calculus have many strange features that are endlessly repeated, Spivak includes many. The worst part is how few calculus books improve on Granville from 1911 which I take as some sort of minimum standard. Buying those books just encourages their authors.

 

[victor.raum]

Get both.

Spivak will definitely be too dense if you're very new, but that doesn't mean it won't be worth it to try reading as far into Spivak as you can. At the same time, the other book can serve as your staple for simple explanations when the Spivak's rigor is making your head spin a bit too much.

 

[Skrew]

Stewart often gives proofs and as a first calculus book I think its perfect. I can't see why certain people dislike the book so much outside of math snobbery.

 

[homology]

There's nothing special about Stewart (or Spivak for that matter). At your first pass there's nothing wrong with just learning the 'rules' for calculation. I suppose it would depend, are you going to spend your career in math or in science? If science then do not start with real analysis, learn how to calculate then learn why you can.

Any intro calc book has some advantages and disadvantages. It'll depend on you of course. I've always liked the idea of https://www.amazon.com/dp/1568811578/?tag=pfamazon01-20 by Frank Morgan which is a fraction of the cost without all the bells and whistles of standard books.

But as I said, it depends on what you'd like to do with calculus

 

[Robert1986]

So, upset, shocked and offended that anyone should dare question the greatness of Stewart, I decided to look at my Stewart Book to find proof that I am correct. (OK, that was a bit dramatic.) But, alas, I find a very serious flaw in the way that Stewart treats a very important subject in calculus: limits. In the body of the text, he never actually gives a definition. There is a definition in an appendix, but there is not even a reference to it in the body of the book. So, yeah, if you have a professor who can teach all the stuff that isn't in Stewart, then may be it is a fine book. However, for a self study thing, I would choose Courant or Spivak.