
#1
Sep2510, 06:16 PM

P: 8

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. 



#2
Sep2510, 07:01 PM

P: 367

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.




#3
Sep2510, 07:58 PM

P: 8





#4
Sep2510, 08:05 PM

P: 8

Apostol vs Spivak
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 mathematiciantobe shall take the later road.




#5
Sep2510, 09:33 PM

P: 326

Sounds like you have a predecided opinion. Perhaps you should allow discussion before making such a strong assertion. :)




#6
Sep2510, 09:41 PM

P: 8





#7
Sep2510, 09:54 PM

P: 326

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.




#8
Sep2510, 09:58 PM

P: 8

Why not? 2. "The way you have developed this conversation is not the optimal way to stimulate debate over the topic." I doubt that. 



#9
Sep2510, 10:31 PM

P: 326

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.




#10
Sep2510, 11:42 PM

P: 418

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 nontrivial 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. 



#11
Sep2610, 12:35 AM

P: 8





#12
Sep2610, 12:42 AM

P: 8





#13
Sep2610, 12:45 AM

P: 326

intuition=higher plane of intelligence
:) 



#14
Sep2610, 12:58 AM

P: 8





#15
Sep2710, 06:35 AM

P: 792

I find this discussion quite interesting. Please could you further explain the differences between Spivak and Apostol.




#16
Sep2710, 04:45 PM

P: 326

I would say that to some degree I am a member of the preintuitionist school. 



#17
Oct210, 07:09 PM

P: 530

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. 


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