Can I benefit from directly using analysis textbooks to self-learn calculus, instead of calculus textbooks?

Question

My purpose is self-learning, neither for exam nor degree courses. My goal is to research dynamic System, theoretically oriented.

1. Question Description: I've been reading calculus books by Weinstein & Marsden, UTM, Springer for weeks. I solved 90% of text, 30%-40% of exercises. UTM seems engineering-oriented (not theoretical/rigorous-oriented), compared with others within series.

Their advantages are: they suitably explain concepts, in clear Structure.

Their disadvantages are: not enough theorems, too many exercises in formula-calculation/real application, too little deep/proof exercises. They total approximately 8000 exercises, 300-400 exercises/chapter, but 80% is simple-formula-calculation/realistic application.

My question: Will I benefit from starting with the analysis textbooks below now, instead of continuing with the aforementioned calculus books? I think so, for 3 reasons:

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

(2) Since the aforementioned books 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), books will still cost several months.

(3) Will the analysis textbooks below also contain needed intuition, calculation skills for calculus? If it's the case that these analysis books train both theory and calculation (compute derivatives/integrals which are useful later such as ODE, PDE), then there'd be no need to read calculus books.

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. Zorich's analysis seems to contain many physical problems, will it works for teaching intuition?

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

Answer 1

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.

Answer 2

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?

Answer 3

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.

Answer 4

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."

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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).

Answer 5

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.