Calculus Why "Transition Books (Apostol, Spivak)" are necessary?

[bacte2013]

Dear Physics Forum friends,

Why so many people recommend Spivak, Apostol, and Courant calculus textbooks, especially as a preparation toward the advanced courses like analysis and abstract algebra? Are they really necessary? I have been studying Apostol's Mathematical Analysis, Rudin's PMA, and Pugh right after completing a computational single-variable calculus course and a self-studying of basic proof methods, and I have been enjoying those texts a lot. It is my understanding that those analysis textbooks cover all contents in Spivak/Apostol/Courant and even more with details and clarity. But some professors recommended me to start with those transition books while other professors recommend to jump directly into analysis and learn thoroughly. Will I be missing something or be in disadvantageous state if I do not finish those transition books?
PK

 

[verty]

Why Spivak, because the questions are progressive and he gives almost no help with them. So it gets you into the mode of puzzling at a question and finding a way to solve it. If that's how you like to learn, it's good.

Incidentally, have a look at Terence Tao's "An Introduction to Measure Theory" (free online). If the style suits you, I see no reason why you would ever want to waste time with a book like Spivak when a more categorical book would be better.

 

[bacte2013]

Thank you for the advice. I just read some portions of Spivak's Calculus but I did not like it that much though...although he provides a lot of motivation and leads readers to a lot of interesting questions, he is very wordy unlike Rudin or Apostol.

Thank you very much for the suggestion! Is Terrence Tao's Analysis I-II required for the measurement theory?

 

[certainly]

The best advice I can give you is that don't worry about problems before they occur. Start your studies in analysis and if you have problems in understanding the material then worry about something that might have been left behind. Personally I think you're quite well equipped for something like measure theory...
Since you seem to like Rudin's style you might want to check out his book "Real and Complex Analysis". He mentions in the preface that the first 7 chapters of his PMA furnish enough prep.
Cheers.

 

[verty]

Thank you for the advice. I just read some portions of Spivak's Calculus but I did not like it that much though...although he provides a lot of motivation and leads readers to a lot of interesting questions, he is very wordy unlike Rudin or Apostol.

I found Apostol wordy, I suppose everyone is different. And Rudin is positively Shakespearean.

Thank you very much for the suggestion! Is Terrence Tao's Analysis I-II required for the measurement theory?

He says you need to have been exposed to undergrad analysis and the Heine-Borel theorem. I certainly don't think he meant in any way to require those books in particular, and I'm sure he wouldn't expect readers of his measure theory book to have read them. Probably he thinks his readers will have had an analysis course and will know it sufficiently well.

He does mention using topological notions toward the end, and measure theory is where they crop up so I'm pretty sure he would define them at least to make explicit how he uses the words, but the sense I get is that it is not meant to be a difficult book.

We must take him at his word: knowledge of Heine-Borel is enough. He will later use the topological concepts but they are easy to understand and when you get there, it'll all make sense.

 

[verty]

Since you seem to like Rudin's style you might want to check out his book "Real and Complex Analysis".

I would recommend against this for the simple reason that a search on these forums will turn up a number of times that people have decided not to use this book. They simply couldn't make head or tail of it and went elsewhere.

 

[bacte2013]

^
Dear Mr. very,

Thank you very much for the all of your advice! Is there a specific reason why Rudin's RCA is not popular? I actually ordered this book on yesterday...

 

[verty]

^
Dear Mr. very,

Thank you very much for the all of your advice! Is there a specific reason why Rudin's RCA is not popular? I actually ordered this book on yesterday...

What does it matter, you bought the book. Soon you will know for yourself whether it suits you and my answer here will be irrelevant. And to others who haven't bought it, you know that I wouldn't recommend it for the reason I gave above, that people have found it difficult. This seems to be a sufficient answer.

 

[certainly]

I wouldn't recommend it for the reason I gave above, that people have found it difficult. This seems to be a sufficient answer.

I'd like to point out that the book still remains famous among mathematicians......

 

[verty]

I'd like to point out that the book still remains famous among mathematicians......

Here is a possible reason why people have struggled with it. :

This year we have been using the 1987's third edition of Walter Rudin's treatise as the main text for a standard first-year graduate sequence on real analysis, backed up by Wheeden/Zygmund's book on Measure and Integral, and the two seem to complement each other quite nicely.
...
One point to keep in mind though, is that Rudin developes the (Lebesgue) measure in a more formal axiomatic direction, instead of in the more concrete, constructive approach. In the constructive approach, one first introduces the "subadditive" outer measure as a set function which is defined on the power set P(X) of a nonempty set X. One then proceeds by showing that the restriction of the domain of the outer measure to a smaller class of subsets of X (a sigma algebra M), obtained via applying the Caratheodory's criterion, results in a "countably additive" set function which is called a measure on (X, M) (The latter is the approach also taken in both H.L. Royden and Wheeden/Zygmund). The formal approach is not very intuitive and is less natural for a beginning graduate student who might not have developed some level of mathematical maturity yet.

 

[certainly]

Well yes, that is true....
if you are not yet mathematically mature, then the book is not for you...

 

[tom26]

If you're self-studying real analysis, make sure you check out the videos of Francis Su's course at Harvey Mudd College. He's a wonderful teacher, and his course covers the first 5 chapters of Rudin.

Personally, I'm not a huge fan of these intermediate books. I found real analysis on the line terribly boring, and I gave up on Spivak a few chapters in. As soon as I started reading chapter 2 of Rudin's PMA, I knew I'd found what I'd been looking for. Another great text is Korner's "A Second Introduction to Analysis" - he covers a lot of higher-dimensional material also - it was only there that I finally understood the importance of the Jacobian.

I think a good example of the clarity the more general approach brings is the triangle equality: in 1D it's this seemingly insignificant formula, but in a general metric space you see it's an unforgetable geometric fact of obvious importance.

 

[bacte2013]

Thank you very much for all advice and arguments. I have been using Rudin's PMA, Apostol's MA, and Pugh's RMA, and they are very fantastic books for self-study! I found that Apostol's MA provide both motivation and details that Rudin's PMA does not have, and no spoon-feeding proof exposition. I tried both Spivak and Apostol's Calculus, but I actually do not like them that much.