Calculus books, intuitive but rigorous?

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Discussion Overview

The discussion revolves around the search for calculus books that provide an intuitive yet rigorous treatment of the subject. Participants share their experiences with various texts, expressing preferences and critiques while seeking recommendations that meet specific criteria, such as self-containment and relevance of applications.

Discussion Character

  • Debate/contested
  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant expresses dissatisfaction with Spivak's book due to poor explanations and illustrations, despite enjoying the problems.
  • Apostol's book is described as dry by one participant, while another suggests that a lively professor may have influenced their positive experience with it.
  • Courant and John's books are recommended by multiple participants, though one notes that the exposition could be more modern.
  • Another participant mentions a book titled "Calculus Unlimited" that fits some of the desired criteria but does not recommend it due to concerns about avoiding limit definitions.
  • There is a discussion about the subjective nature of intuition in learning, with one participant noting that what works for one may not work for another.
  • Concerns are raised about the lack of professional reviews for certain books, with a suggestion that personal preference should guide choices instead.
  • One participant questions the coverage of implicit differentiation in a recommended book, leading to uncertainty about its content.
  • Another participant expresses a desire for more applications in Spivak's book, comparing it to Lax's book.
  • A request for information about "Advanced Calculus: A Differential Forms Approach" is made, seeking insights into its level and prerequisites.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the ideal calculus book, with multiple competing views and preferences expressed regarding different texts. The discussion remains unresolved regarding the best approach to learning calculus.

Contextual Notes

Participants express varying levels of satisfaction with different books, highlighting the subjective nature of learning preferences and the importance of teaching style. There are mentions of specific content concerns, such as the treatment of applications and implicit differentiation, but no definitive conclusions are drawn.

whyevengothere
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I've tried to learn calculus many times from many books,I've come to the conclusion that there is no ideal book on this subject.

I've read Spivak's book,and greatly enjoyed its problems but I felt unstatisfied by the explanations and the illustration were very poor ,and the only chapter on application is the one on planetry motion(which is beautiful).

As for Apostol's book ,I've read only a few chapters of it,and the only thing I can say is that it is indeed very dry.

I've also read some portions of MIT's free calculus book by Strang and it's really great ,but it doesn't preapares you for more mathematical treatments of analysis (there's no discussion of least upper bounds and other things).

Can anyone comment on these books ,especially the last two which I haven't read too much of, I would like to know if there is any calculus (or anything else) book that fits the following criteria:

-riogorous but intuitive treatment with a geometric flavor(non-axiomatic approach if possible).

-self-contained.

-contains the most relevant applications and makes use of them.
 
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I'm not sure what to suggest. When I saw the title I was going to recommend Apostol, but you found that to be too boring. I had an incredibly lively and wonderful professor for an honors calc 1+2 class that used that book, so it's possible that it was the addition of the amazing professor that made things seem so lively. That said, I do recall liking the book quite a lot too and being a bit disappointed when we switched to another offer for honors calc 3 in the spring (although it was still a great course). However, once again, it's entirely possible that had I not had that professor I would have been less excited by Apostol. That was one of the greatest course sequences I've ever taken. Perhaps you should give Apostol another chance?
 
porcupine137 said:
I'm not sure what to suggest. When I saw the title I was going to recommend Apostol, but you found that to be too boring. I had an incredibly lively and wonderful professor for an honors calc 1+2 class that used that book, so it's possible that it was the addition of the amazing professor that made things seem so lively. That said, I do recall liking the book quite a lot too and being a bit disappointed when we switched to another offer for honors calc 3 in the spring (although it was still a great course). However, once again, it's entirely possible that had I not had that professor I would have been less excited by Apostol. That was one of the greatest course sequences I've ever taken. Perhaps you should give Apostol another chance?
I'll try to read more of Apostol's book,but what do you like about expilicitly?
 
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whyevengothere said:
I would like to know if there is any calculus (or anything else) book that fits the following criteria:

-riogorous but intuitive treatment with a geometric flavor(non-axiomatic approach if possible).

-self-contained.

-contains the most relevant applications and makes use of them.

The only book I know of that is like this is Calculus Unlimited, you can find an online copy on the author's website. That said, I don't think it is a good idea to avoid the usual limit definitions. So I don't actually recommend this book.
 
whyevengothere said:
intuitive treatment

Your intuition is different from my intuition.

(IOW: what works for others, doesn't have to work for you, so there is no guarantee advice given will work).
 
  • #10
Borek said:
Your intuition is different from my intuition.

(IOW: what works for others, doesn't have to work for you, so there is no guarantee advice given will work).

Then,the opposite of formal presentation,with abstraction kept at a minimum.
 
  • #11
whyevengothere said:
I like it too,but since I'm not an expert,I tried looking for reviews by professionals,but couldn't find any online,any help?


Well, you can't go wrong with any book written by Peter Lax. And why is it so important to find reviews by professionals? If you like it, that should be enough, right?
 
  • #12
Xiuh said:
Well, you can't go wrong with any book written by Peter Lax. And why is it so important to find reviews by professionals? If you like it, that should be enough, right?
I look into it on amazon and there's nothing on implicit differentiation,is that right?
 
  • #13
whyevengothere said:
I look into it on amazon and there's nothing on implicit differentiation,is that right?

I'm not really sure, but if I remember correctly the book has nothing on implicit derivatives.
 
  • #14
Xiuh said:
I'm not really sure, but if I remember correctly the book has nothing on implicit derivatives.

but aren't the problems too easy?
 
  • #15
If Spivak's book had half the application in Lax's book ,it would be the perfect calculus book,I have the third edition ,does the fourth have more applications?
 
  • #16
For multivariable calculus,I found
Advanced Calculus: A Differential Forms Approach
Harold M. Edward
I would like to know if anyone know anything about it, what's its level?What does it require? Anything?
 

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