# Loomis "Advanced Calculus" and Baby Rudin

1. Jun 10, 2014

### Thinker301

Hello everybody!

I was wondering how the two compare?

Thank you everybody :)

2. Jun 12, 2014

### GeneralOJB

My money is on Rudin.

3. Jun 12, 2014

### verty

Wow, I think you are jumping into a potential world of pain here, although perhaps that is what you want. You may however want something just as concise but easier as an intermediate step to this Sternberg/Loomis book.

This book looks very concise but less advanced, slow and steady wins the race:

Last edited by a moderator: May 6, 2017
4. Jun 12, 2014

### Thinker301

Oh no I just wanted to know how they compare to each other.

5. Jun 13, 2014

### mathwonk

look at the math textbooks thread for discussion of rudin. there seems to be no listing there for loomis sternberg although that surprises me. i must admit though i am always tempted to suggest you just look at them yourself. i.e. if you have access to them I strongly recommend you just look at them.

however, since you ask, as a little info, I suggest they have almost no similarity at all. rudin is a hard core analyst who delights in making analysis look elegant, explicit, and almost totally unmotivated. As I age more and more, I think of his books more and more as just masochistic exercises. If you are very bright they may just seem challenging, but if you are just hoping to learn, they do not help much.

Loomis is similar in one way, he values elegance above all else, but he is much more abstract than rudin, and seems actually trying to make his book the most elegant and most abstract book in existence. His book is beautiful, and in some ways elementary, in that he assumes very little knowledge, but the level of abstraction and sophistication is very high. Neither of these books is recommended to learn from, at least by me. They both amount to an exercise in showing off by the authors.

I kept loomis on my shelf as a historically interesting and impressive work, (and the sections by sternberg are more explanatory), but I finally just got rid of Rudin altogether, gave it away, as pretty much a worthless book, with nothing I want to read.

To repeat myself, these are books that the authors are proud of, but for non pedagogical reasons, i.e. they show he is smarter than we readers are, an unworthy goal in my opinion for an author.

A good explanation should reveal where the ideas came from, why they are interesting, why they are natural approaches to the topic, and how they may be seen as easy, almost predictable. Neither of these books has that aim nor achieves it. To me, these are authors who want to impress rather than educate the reader.

Still, they ARE experts, and thus one has something to learn from them, so if the books speak to you, go ahead and learn from them. But if you are an average reader, this result is not likely to occur. I.e. even though they have something to teach, they do not try very hard to teach it, and that job is left almost wholly to the student.

Again, Sternberg is much more down to earth and helpful than Loomis, so the second half of their book, the half apparently written by Sternberg, may be much more accessible.

Actually Loomis is so skillful that his book also is accessible, it is just that after reading it one will not have any skill at all with using the concepts. One will get a clear idea of the theoretical side from his beautiful explanations, but one will have almost no grasp of the practical side. In the case of Rudin, one is unlikely even to understand the theory.

If you want to understand calculus of differential forms, read spivaks calculus on manifolds, not rudin. and if you want to understand measure and integration, read berberian or wheeden and zygmund, not rudin.

if you want to understand metric spaces, maybe read mackey's complex variables, or dieudonne's foundations of modern analysis (not easy either), rather than rudin, or maybe simmons. maybe one can benefit from working rudin's problems more than reading his explanations.

but again, read them yourself, and see whether they speak to you. don't be bound by a grumpy old man's views. You are young and intelligent. Read them and make up your own mind!

Last edited: Jun 13, 2014
6. Jun 13, 2014

### mathwonk

I guess that rant was about their style, whereas you probably care about content. In content rudin is far more elementary, beginning with a careful and tedious development of just how to construct the real numbers via the method of "dedekind cuts". I.e. the real line has the property that removing any point separates it into two unbounded disjoint open intervals. The idea behind dedekind's cuts is to postulate the converse, that for any pair of non empty unbounded disjoint intervals in the real line there must be at least one point in between them which has been omitted.

Put another way, if we remove any unbounded open interval from the real line, the complement is a closed unbounded interval, i.e. it must have an endpoint. That interval is thought of as representing the real number which forms its endpoint. More generally the decomposition of the line into two intervals is thought of as representing the real number forming the endpoint of the closed one.

Now to express real numbers purely in terms of rationals, we just speak instead of a decomposition of the rationals into two disjoint intervals. Then a real number is defined to be a decomposition of the rationals into two unbounded intervals. If one of these intervals is closed the associated real number is the rational endpoint. If not, the associated real number is the unique real number lying between both rational intervals.

Well, I told you it is tedious. Rudin then develops all of one variable calculus in a completely precise way, and goes on to several variables i believe. He ends with a brief development of measure theory and the associated integration theory due essentially to Lebesgue.

Loomis and Sternberg, pretty much assume you have had elementary calculus of one variable, including Riemann integration, and develop first a sophisticated theory of differential calculus in normed complete vector spaces (Banach spaces) and go on to develop a less sophisticated integration theory than Lebesgue's, rather an n dimensional version called the theory of "content" and the associated Riemann like integration theory.

Probably both books do the theory of implicit and inverse functions, certainly Loomis does.

Then Loomis - Sternberg go into manifold theory, and differential forms, more geometrically I think than Rudin does, and afterwards give applications to potential theory and classical mechanics.

The overlap is so little, in style if not precisely in content, that one could easily imagine first reading Rudin and then Loomis Sternberg and not being bored nor seeing much repetition at all in the second one.

But as I said, one gets a somewhat special view of these subjects from these books, not ones I really recommend for first encounters with the material. Loomis and Sternberg are a bit too abstract, so that you may not really get a feel for the material, and although Rudin is not too abstract and his material is standard analysis as beloved by many analysts, his writing is just rather hard for most students to benefit fully from, because he never shows how one might think of the arguments oneself.

But if you are someone for whom Rudin works, you might read him first, working lots of problems, and L-S second.

7. Jun 13, 2014

### robphy

8. Jun 13, 2014

### Thinker301

@robphy

Thank you! I actually first encountered L&S from the pdf, I then looked for what other thought of the book.

@Mathwonk

Thank you! For your second post first. I see Rudin and L&S analogous to what SV and MV Calculus are to each other.

What would you suggest from someone who just really loves a challenge?

Also I want to say I really appreciate everything you said, it was all really helpful. I just don't really have any questions or additive comments to most of it. But I really do appreciate it! Thank you!

Last edited: Jun 13, 2014
9. Jun 13, 2014

### mathwonk

"What would you suggest from someone who just really loves a challenge?"

read rudin, do the problems and then read L-S. but first it helps if you have done computational calculus, such as from edwards and penney or stewart.

10. Jun 13, 2014

### Thinker301

Sounds like a plan. I was actually thinking

Spivak -> Apostol 2 -> Rudin -> L&S

but does

Computational Calculus -> Rudin -> L&S work as well?

11. Jun 13, 2014

### mathwonk

I suspect you really don't need rudin after spivak and apostol 2. and maybe not L-S either. I.e. at that point you know calculus, and might as well move on up to more advanced stuff.

I would look but i admit i even gave away my copies of sopivak and apostol. not out of disrespect but to upgrade the undergraduate math major library at my school.

of course spivak is online, but i don't know about apostol. i'll look.

and oh yes, L-S also has a chapter on differential equations and a very strange chapter one that tips you off how wacko the 60's were: it is an elementary chapter on logic and sets and functions, including equivalence classes.

now who does not know what an equivalence class is, and yet is ready in the next chapter or two to tackle banach space calculus?> the brilliant harvard freshman of 1960, that;s who.

12. Jun 13, 2014

### mathwonk

ok i looked at apostol online and i agree that it might make sense to read L-S after it, because it is about finite dimensional calculus and surface integrals in 2 and 3 dimensions, whereas L-S does the n dimensional cases.

i might suggest however just doing apostol 1 and 2 instead of spivak. and i would still probably omit rudin.

so i suggest apostol 1, 2, and then L-S.

Have you looked at Dieudonne'? Foundations of modern analysis?

it would come after either apostol 2 or L-S. It is a very remarkable book.

I still have my copy from 1960 or so.

these prices on it are unbelievably cheap: (it is worth at least \$50.)

http://www.abebooks.com/servlet/SearchResults?an=dieudonne&sts=t&tn=foundations+of+modern+analysis

13. Jun 14, 2014

### Thinker301

Thanks!

Just wondering, why Apostol over Spivak?

Also I will certainly check it out.

14. Jun 14, 2014

### mathwonk

well they are both great. Spivak is more fun, Apostol is maybe more detailed and "scholarly".

I just thought if you are choosing Apostol for part 2 you might be better off using it also for part 1,

on the other hand maybe a change of pace is better. Not a big deal. Both books were written in the 60's for very bright but unsophisticated students. Both very well written. You cannot go wrong with either one.

And fun is not to be despised.

You really owe it to yourself to actually look closely at them and make the decision yourself. Whatever speaks to you is the best book for you.

I myself learned single variable calc in 2 stages, 1st I graded an honors course using spivak, then second, I taught a course out of spivak and read courant at the same time. so my sources were spivak and courant.

'then i looked at apostol and was very impressed.

as to several variable calc, lets see, i read and taught from spivak again, calc on manifolds, and also backed it up with courant part 2, and also dieudonne, and lang, analysis 1 and 2, and maybe one of my favorites is fleming, calc of several variables.

so spivak has been my book of choice for learning, because he is so clear, but i really like apostol too, i just did not have access to that book at the time.

Last edited: Jun 14, 2014
15. Jun 14, 2014

### Thinker301

That makes sense. I will certainly pick Apostol 1 if I am going to pick Apostol 2. I really enjoyed Apostol's style while I briefly had the book. It felt a lot more precise than Spivak. But I might have to go Spivak -> Advanced Calculus by Folland.

Can I ask you a question which is slightly off-topic but will determine the books I use?

16. Jun 14, 2014

### mathwonk

but let me emphasize again, very few of us can just plow through a given book, no matter how recommended it is. we need to choose a book that we can learn from. for most of us this is a different book for every different topic. we learn differentiation from one book, riemann integration from another, lebesgue integration from another, inverse function theorem from another, but spivak is very very clear on whatever he undertakes, as is apostol.

analysis/calculus books i have learned from: spivak, courant, dieudonne',..,.,. gosh is that it? maybe also fleming, riesz-nagy, berberian, lang,...but your l;ist will be different, and then you will be giving advice on what books you benefited from. take a leap (of faith, not in the lake).

17. Jun 14, 2014

### Thinker301

Hmm I see, its not really about the books but the subjects and I should just look for somewhere I can learn more about that topic from

That's incredibly illuminating.

Thank you !

Last edited: Jun 14, 2014
18. Jun 14, 2014

### verty

By all means get L&S, it sounds like you want this type of book. I thought it might help to see multivariable calculus in an easier book first, that was the only reason for my comment. One thing, the computational multivariable books that I have seen I have not liked, for me they have not been conceptual enough. For example, directional gradients are often presented as a formula: $\nabla f \cdot \hat{n}$. But basis vectors are to some degree arbitrary; this formula should work with any basis vectors and it should also work in one dimension as well. So there is some kind of deeper concept here that is not often explained.

So my original suggestion was intended as a conceptual first exposure to the multivariable material. Of course I think Apostol 2 is a very good choice but it is a very expensive book.

19. Jun 14, 2014

### Thinker301

Thanks verty. What do you think of Advanced Calculus by Folland instead of Apostol 2?

20. Jun 14, 2014

### verty

From the limited information I could attain, this book seems to cover n-dimensional calculus on Euclidean spaces. I'm learning towards thinking that this is how multivariable calculus should be learned. I'm not 100% convinced but connections are good and if it does everything in a unified way, what a pleasure.

One or two reviews I read suggest that there are not many examples, I'm guessing this is a do-almost-every-problem type book which is usually a good thing, it forces one to put the effort in. The reviews are not glowing, what comes across though is that it is "functional" at what it wants to achieve. But if it functions to teach calculus in this way, yeah, I can't see a serious problem with that.

I can't see inside the book but it looks usable, probably a book I myself would use in your position. The errata is available in PDF form, so I think a good choice given the low price that one can get it at.

As to how it compares to Apostol, I know not a lot about Apostol 2 but I see that it contains other content like linear algebra and linear differential equations. This is stuff you can get in other books, and Folland does include Fourier series. So it's a less linear book than Apostol 2, which I suppose makes sense because Folland also wrote a real analysis book. Different content, similar level is my best guess.