Heard many people say that there are three good cal textbooks: the ones by Apostol, Spivak, and Courant. I own Apostol's and Spivak's. The major difference between the two is the degree of rigor and logical order, in which Apostol's apparently beats Spivak's, although Spivak's is far better than most other cal books in these aspects. Want demonstration? Just go to the first section of each book. Check out Spivak's chapter on properties of numbers and Apostol's chapter on real set and the field axioms. Conclusion: Really serious math students and future mathematicians shall pursue Apostol for their elementary calculus education.
I don't know which book is better but Spivak proves almost every theorem in the book The exceptions beings being the chapters on Complex Analysis and the proof that each integer factors uniquely and each fraction has a partial fraction decomposition. He also leaves out some stuff in the exersize that proves uniqueness for solutions of linear constant coefficient differential equations of any order. I'm say that was pretty riguros. Also the problems are extremely difficult for a calculus book - harder imo then those in Artin's algebra and Arnold's ODE for example.
The problem is in the foundational part, Spivak's assumes something but doesn't explicitly state it. For example, he assumes some elementary algebraic rules, such as if a=b, a+c=b+c, and then use them to proceed his proof. That's not rigorous enough compared with Apsotol's. As to the logical order, Spivak shall not state something like "since 0*b=0" first and let the readers regard it as his another tacit assumption while "prove" that using the "property" he lists out later. The order is just not right. On the other hand, Apostol lists out all the axioms that will be used for proof first and then lists out the theorems to prove later. For example, 0*a=a*0=0 is listed as a theorem to prove.
One more thing: Spivak likes to lists out a lot of theorems and then proves them directly in his conversational/informal texts while Apostol highlights them as theorems in another section and then lets the readers to prove most of them by themselves. Really serious mathematician-to-be shall take the later road.
Sounds like you have a pre-decided opinion. Perhaps you should allow discussion before making such a strong assertion. :)
I do have pre-decided opinion (who doesn't?). But that does not mean no discussions are allowed. Do you see in every discussion people without "pre-decided opinions"?
I'm simply stating that it is best to keep an open mind, for the sake of the conversation. If you are already certain of something, why start a thread on it? The way you have developed this conversation is not the optimal way to stimulate debate over the topic.
1. "If you are already certain of something, why start a thread on it?" Why not? 2. "The way you have developed this conversation is not the optimal way to stimulate debate over the topic." I doubt that.
Getting back on topic, from what I've read (I am not yet at the level of calculus), Spivak is a much better introduction, and Apostol is more rigorous...Spivak is likely to be much more enthralling, and thus is probably best as a first course.
What exactly does rigorous mean to you? Spivak doesn't get as bogged down in certain fundamentals, but so what? The books cover different things and in different ways. Apostol chose to cover some foundations of natural numbers in a more thorough manner, but this does not make it more rigorous. The axiomatic method employed by Spivak is just as valid. Spivak often leaves small gaps and assumes prior knowledge of things like the integers, but so does most serious math books. When you read a graduate math book it may state "because [itex]\pi_1(\mathbb{S}) = \mathbb{Z}[/itex] we have ..." or something else. The proof of this fact is not non-trivial or unimportant (in fact many algebraic topology books have this very result as the main result in one of their early chapters), but the author just choose to assume it as a prerequisite. Personally I feel Spivak is much closer to the style of graduate text books, and while he is often not as precise and thorough this style promotes more critical thought and arguments guided by intuition rather than symbol manipulation. This is of course just my opinion, and both books are perfectly good as introductions to calculus.
Graduate texts are always dry and dense and their style are not like Spivak's, conversational and informal. You're right: Spivak's is not that precise. So it may better suit college freshmen. College freshmen (excluding serious and budding mathematician-to-be) always need motivation on "intuition" and are always "bogged down" by cool, uninspiring, and dry "symbol manipulation", aren't they?
Are you quoting Henri Poincaré's definition of intuition? Do you think Klein possess "higher plane of intelligence" than Hermite? xD
I find this discussion quite interesting. Please could you further explain the differences between Spivak and Apostol.
If you buy this book you should have no trouble with either Spivak or Apostol, or both! Here is a detailed description of the book. From looking at both books I wouldn't be happy unless I read both, to me there is simply no question about it. Apostol covers a lot of Linear Algebra and differential equations, approaches calculus more historically and balances computational calculations with proofs. Spivak throws you in the deep end by leaving most of the substance of the chapter to the questions but has a way of making it work - provided you're prepared.