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Calculus textbook using now[1]: Calculus I, Weinstein&Marsden, UTM, Springer

Question Description: I've been reading book[1] for weeks, 90% of text, 30%-40% of exercises solved. It's not bad, but for the following:

Advantages: (a)Suitably explained for concepts (b)Clear Structure

Disadvantages: (a)Not contain enough theorems (b)Too many exercises in formula-calculation/real application (c)Too little deep/proof exercises (d) Approximately 8000 exercises in total, 300-400/chapter, but 80% is simple-formula-calculation/realistic application.

My Opinion: Will it be more beneficial to start using analysis textbooks now instead of this calculus book ? For 3 reasons:

(1). Most good EU bachelor in maths, they use analysis directly in first semester instead of calculus. (e.g. Bonn University/ETHz)

(2). Since book[1] contains too many exercises of formula-using/real application ones but not deep/proof, if I continue to work with it (solve all exercises/ second time reading), book[1] will still cost several months, I doubt if it's beneficial compared with directly starting analysis.

(3). Will Analysis textbooks(e.g. book[4][5]) also contain needed calculus?(intuition/calculation skills) If it's the case, such analysis books would do both to train modern theory and calculation skills( compute derivatives/integrals which are useful later such as ODE,PDE), then there'd be no need to read calculus any more.

Future Goal: Research in Dynamic System theoretically oriented.

Note: Though [1] is UTM, but it seems engineering-oriented(not theoretical/rigorous-oriented) compared with others within series.

[2]Rose, Elementary Analysis, UTM, Springer.

[3]Serge Lang, A First Course in Calculus/Calculus of Several Variables, UTM, Springer(Even though it's still calculus, but Lang's book is more abstract-oriented)

[4]Zorich, Analysis, Universitext, Springer(As @nbubis said, analysis needs intuition behind, from the content, it seems Zorich's analysis contains many physical problems, will it works for that ?)

[5]Courant, Introduction to Calculus and Analysis I&II, Springer

Desirable answer: Advices, Discussions

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They are very different subjects. You will need the background experience in calculation. –  André Nicolas Jun 14 '12 at 19:36
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Since you mention Bonn and the ETH, there is a chance that you can read mathematical German. In that case my recommendation would be to study Königsberger's book in two volumes Analysis 1 and Analysis 2 (Springer).It is meant for German students in the first semesters of universities. (The first volume is devoted to analysis in one real variable) . Königsberger manages to keep a perfect balance between clarity and absolute rigor. The examples are neither trivial nor difficult and so are the exercises (in volume 1 all are solved). The illustrations are simple, austere and evocative. –  Georges Elencwajg Jun 14 '12 at 19:48
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By the way, I agree with you that books with too many exercises are bad. I have silently thought so for a long time, but strangely it is the first time I see this mentioned publicly . ( But I'm not going to start a discussion on that subject !) –  Georges Elencwajg Jun 14 '12 at 20:06
    
@GeorgesElencwajg is it ok to start Königsberger now to replace current Calculus(Marsden,UTM-Springer) ? Because as Magidin said calculation skills/techniques are also important and useful, but I'm a little less patient for current calculus-book containing too many formula-using/application exercises without enough theories/proofs. So my question is , in general, is it also possible to get calculation skill/technique from analysis book like Königsberger ? If it's the case, then I only need to use Königsberger to enjoy the theorem and also not to lose calculation training ? –  Xingdong Zuo Jun 15 '12 at 19:53
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Dear Xingdong, Analysis 1 has 17 chapters and each chapter has about 15 exercises as an average.You will need skills and technique to solve them because they are esentially calculations. However these calculations are interesting ,challenging and yet not excessively difficult. It is more instructive to go through one of them than blindingly solve ten boring exercises that are infinitesimal variations of each other. And, as I wrote, the problems are solved at the end of the book. Anyway, try Königsberger: you can always come back to your textbook if you fear you are forgetting your skills. –  Georges Elencwajg Jun 15 '12 at 20:26

5 Answers 5

up vote 3 down vote accepted

I would agree. I had taken some non-proof high school Calculus, so I am not sure if my experience would be completely similar to someone who wants to go straight into analysis.

I think someone with no background in calculus could read something like Principle of Mathmatical Analysis by Walter Rudin with no great difficulty. I was able to read this book without any proof experience. In fact, the beginning of Rudin are basic metric space topology and least upper bound property results which I feel are more suitable materials for learning proofs than the more tedious proofs of theorems about derivatives and integrals found in a Calculus book. Most analysis text like Rudin will eventually cover the fundamental results of Calculus like derivatives, integrals, means values theorem, Taylor Theorem, etc. However, as you mentioned there less are emphasis on on example and calculations (which has caused me some headaches later in my studies).

So I would say if you are more interested in studying pure mathematics in the future a real analysis text like Rudin or Pugn would be a good introduction to how to do proofs. Also a Calculus book by Spivak is also a good place to learn how to do proofs and calculus as well. If you are more interested in science, applied math, you may want to take a look in a Calculus book that emphasizes Calculations.

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At my undergraduate institution (Facultad de Ciencias, UNAM, Mexico), for a while in the mid-to-late 70s, several professors in the Calculus sequence (four courses: Differential single-variable (Calc I), Integral single-variable (Calc II), Differential multi-variable (Calc III), Integral multi-variable (Calc IV)) decided to use Hasse's analysis textbook instead of a calculus textbook. It was more of a "baby analysis" than a calculus course.

Now, this was done only in courses that were being taught to Math, Actuarial Sciences, and Physics majors (and a Math major takes nothing but math courses, for instance).

It did not go well. Students didn't learn analysis very well, and they certainly did not learn the calculus skills they needed very well. The Physics department, in particular, went up in arms because the Physics majors were coming out of these courses unable to actually compute integrals and derivatives, or use them to solve specific physics problems. Same problem with the actuarial scientists. The math majors fared a little better, but mainly because the same people who were doing this were the people who were also teaching the analysis courses in the junior and senior years; but those that went on to take analysis from other people didn't do so well. In addition, the failure rate for these courses was extremely high. (Failure rate in the Calculus sequence has always been way too high there, but it got much worse).

Most professors switched back to calculus books and to not do baby analysis. By the mid-80s, almost nobody was using Hasse's book or teaching "mini-analysis."


If a student has had a good enough calculus course in High School, then it is likely that a baby analysis course might indeed be beneficial, building on the bases that calculus can help set. This could very well be the case in the EU; it's not the case in the US. (In Mexico, nominally, students in the Math/Physics/Engineering track were taking a year of Calculus as seniors in High School, but obviously not good enough).

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Since I've read book[1] for once, I've got some calculation skills, but so far still not extreme solid. For Zorich[4] , it's analysis containing physical applications, will it be better choice to do both rigorous proof/theorem and also calculation skills. Since if I stay in book[1], 8000 exercises would still take several months maybe, but by then, theory knowledge/skill maybe not improve much compared with starting analysis ? –  Xingdong Zuo Jun 14 '12 at 19:44
    
@Xingdong Zuo: First, I would not expect anyone to do all problems in a calculus textbook. Second, if you will be doing anything that may require you to actually use calculus (numerical analysis, differential equations, mathematical modeling, probability, statistics), then you need calculus, not just analysis. If you want "realistic applications", you can't really go to analysis books, you need to go to modelling books. Solid theorem/proofs can, and probably should, wait until later. –  Arturo Magidin Jun 14 '12 at 19:47
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Just as a data point, since the original poster brought up Bonn university: German high schools have mandatory courses in calculus in 11th/12th grade. –  Johannes Kloos Jun 14 '12 at 20:21
    
@ArturoMagidin For calculus, do you advise Courant's two volumes "Introduction to Calculus and Analysis" will suit these need?Because from the content, it seems contains many physical applications which will cover both intuition and calculation-skill. Since it's high-reputed and another Serge Lang two titles in Calculus.I find this two to be said high-quality after searching. Do you advise these two would be enough for solid skills in calculus before analysis? (Maybe third one as many people suggest Spivak.) Since there're too many basic calculus text which impossible to read every good ones. –  Xingdong Zuo Jun 16 '12 at 7:30
    
@XingdongZuo: I am not familiar with Lang's book; Spivak is a bit on the theoretical side, Courant is a bit better (in my humble opinion) towards developing calculation skills (with, to some extent, should be an important goal in Calcul* us. What I've heard of Lang's books, though, is not very flattering (Story goes he was driving to a conference with some people and claimed calculus textbooks were easy to write; another passenger claimed it was not so, Lang replied he could do it on a weekend, did so, and that "it shows".) –  Arturo Magidin Jun 16 '12 at 18:47

I think it's worth while going over a calculus book before reading a book on real analysis, since:

  • It's much easier to understand the concepts rigorously after you already have some intuitive idea of what is going on.
  • Learning (even generally) about the applications of calculus, helps one understand the reasons behind the various axioms and definitions.

Most people actually study the topic in this order, after first being introduced to basic calculus in high school, an only then studying real analysis at university.

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For dynamic systems, real analysis is not sufficient but necessary. If you don't understand former sentence go and do a proof based book. http://www.people.vcu.edu/~rhammack/BookOfProof/

This book is gold and helped me a lot, before reading this I tried to deal with munkres topology chapter 1, and simply I got buffled. After reading this book everything is more clear to me.

The difficulty of analysis is not because it is analysis, but just because it is rigorous you need to be comfortable with proofs and basic logic.

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I think it all depends on your own personal wishes and goals. In my opinion, books called "analysis", as opposed to "calculus" named, aim more for people who is going to need heavy analytical weaponry: mathematicians, physics, some (very few!) engineers, whereas calculus is, I think, more suitable for people pursuing scientific careers without the need of too heavy stuff: chemistry, biology, etc.

Of course, upon choosing graduate studies, things can change: my best friends in university were physicists and in graduate school they damned their lack of good background in higher linear algebra and, say Lie groups...

Again, what's your goal? Engineering, sciences...or are you more heavily mathematics-bound?

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For the future, I hope to do Dynamical System related works, especially its core theories in Mathematical Neuroscience, but more math-oriented. The reason I asked above question is because if I continue book [1], it will still take me very long time to finish. But for intuition or application, if concurrently do analysis textbook with some classic mechanics, may it also get physical intuition behind analysis' theorem/def ? Maybe my problem? Since i'm not very patient for book[1] formula-calculate/application exercises, but desiring more rigorous/theorem and proof problems. –  Xingdong Zuo Jun 14 '12 at 19:19
    
@Xingdong Then perhaps you should go into analysis, though there are quite some good books in Calculus, like Spivak's or Apostol's. Good luck. –  DonAntonio Jun 14 '12 at 19:29
    
"my best friends in university were physics" - did you mean physicists? :) –  Ben Millwood Jun 14 '12 at 19:55
    
Indeed. Edited. Thanks –  DonAntonio Jun 14 '12 at 21:07

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