# Apostol vs Spivak: A Calculus Textbook Comparison

• ivanos
In summary: They are a demonstration of the validity of a theorem. Apostol does not leave these gaps and proofs are explicitly stated.The difference I see is that Spivak assumes that the reader already knows these things and that they will be able to fill in the gaps. For example, in the chapter on real numbers, he starts with the axioms and proves theorems with those axioms, but does not say "by the way, to prove this theorem, we need the fact that \sqrt{a^2+b^2}=c^2", he assumes that the reader knows this. Apostol, on the other hand, states the theorem outright as the first theorem in
ivanos
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.

deluks917 said:
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 let's 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. :)

G037H3 said:
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.

G037H3 said:
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 $\pi_1(\mathbb{S}) = \mathbb{Z}$ 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 textbooks, 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.

rasmhop said:
What exactly does rigorous mean to you?

I've provided examples in my response to deluks917.

rasmhop said:
Personally I feel Spivak is much closer to the style of graduate textbooks, 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?

intuition=higher plane of intelligence

:)

G037H3 said:
intuition=higher plane of intelligence
:)

Are you quoting Henri Poincaré's definition of intuition? Do you think Klein possesses "higher plane of intelligence" than Hermite? xD

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

ivanos said:
Are you quoting Henri Poincaré's definition of intuition? Do you think Klein possesses "higher plane of intelligence" than Hermite? xD

I would say that to some degree I am a member of the pre-intuitionist school.

G037H3 said:
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.

If you buy https://www.amazon.com/dp/007001390X/?tag=pfamazon01-20 book you should have no trouble with either Spivak or Apostol,
or both! http://myrtlehocklemeier.blogspot.com/2008/02/tater-says-that-cletus-knows-him-some.html 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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## 1. What are the main differences between Apostol and Spivak's calculus textbooks?

The main difference between Apostol and Spivak's calculus textbooks is their approach to teaching calculus. Apostol's textbook is more traditional and focuses on a step-by-step approach to solving problems, while Spivak's textbook is more abstract and emphasizes conceptual understanding over memorization of formulas.

## 2. Which textbook is better for beginners in calculus?

For beginners in calculus, Apostol's textbook may be a better choice as it provides a more structured and straightforward approach to learning the subject. However, Spivak's textbook can also be beneficial for those who are looking for a deeper understanding of calculus.

## 3. Which textbook is more challenging?

Spivak's textbook is generally considered more challenging due to its abstract approach and emphasis on conceptual understanding. However, both textbooks cover the same material and can be challenging in their own ways.

## 4. Which textbook is more suitable for self-study?

Both Apostol and Spivak's textbooks can be used for self-study, but it ultimately depends on the individual's learning style and preferences. Apostol's textbook may be more suitable for those who prefer a structured approach, while Spivak's textbook may be better for those who enjoy a more abstract and challenging approach.

## 5. Which textbook is more commonly used in universities?

Both Apostol and Spivak's textbooks are widely used in universities, and it ultimately depends on the professor's preference. However, Apostol's textbook may be more commonly used in introductory calculus courses, while Spivak's textbook may be used in more advanced courses or for students who want to pursue a deeper understanding of calculus.

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