# Advanced calculus Textbook

1. Jan 30, 2007

### ayas

Hello,
I am currently taking an advanced calculus class and the textbook is "Advanced Calulus" By Folland.
This textbook is interesting because it goes deep into every subject treated, but at the same time I HATE it because it is very thin compared to the material treated: ie everything is treated but very briefly, few examples, few or no graphical illustrations, not enough exercises, no worked out answers for the exercises(mere numerical answers or nothing when it's supposed to be a proof) and last but not least: not enough intuitive motivation.
So here is my question: can you suggest me advanced calculus textbooks that you know that would have as many as possible of opposites of what I just listed.
This is for a 2-semester class at the sophomore level in calculus of several variables that follows the first year calculus which is restricted to students "with strong background in math".
(I would love to find a 2nd year calculus textbook whose format is like a first year calculus textbook: big and thick, plenty of exercises, intuitive motivations, illustrations,...)
I did well last year and I'm not doing so well this year, and the biggest change was the book I believe. (because our first year calculus was already intense! delta/epsilon, and so on)

Last edited: Jan 31, 2007
2. Jan 30, 2007

### mathwonk

there used to be one by crowell, williamson, and trotter.

Last edited: Jan 31, 2007
3. Jan 30, 2007

### eccefeles

"Calculus: Single and Multivariable" by Hughes-Hallett et al?

(I have no idea what is involved in the sophomore level. It's a first/second year textbook at my uni.)

4. Jan 30, 2007

### Pyrrhus

If you want a good "Advanced Calculus" book then go for Advanced Calculus by Wilfred Kaplan is one of the most insightful books i've ever read on Real and Complex Analysis without being full featured Analysis book.

5. Jan 30, 2007

### mathwonk

standard books from the mid 20th century include kaplan, widder, franklin, buck, taylor, osgood, wilson,

but i kind of liked marsden and tromba.

of course courant vol 2 is superb, and also apostol vol 2.

Vector Calculus: Second Edition
Marsden, Jerrold E. and Anthony J. Tromba
Bookseller: Book Baron Anaheim
(Anaheim, CA, U.S.A.) Price: US$10.00 [Convert Currency] Quantity: 1 Shipping within U.S.A.: US$ 4.00
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Book Description: W.H. Freeman & Co. (1981), New York, 1981. Book Condition: Very Good+. 2nd Edition. Hardcover No DJ- glossy black & white pictorial boards. Usual shelfwear. Bookseller Inventory # 370299

Last edited: Jan 30, 2007
6. Jan 31, 2007

### mathwonk

fopc, how are you finding this book? It seems quite advanced to me, and more suitable for a graduate course than an undergraduate one.

I.e. I think it treats differentiation in infinite dimensions at the same time as finite dimensions.

OP asked for one with explanations, motivation, examples, problems, not just pure high powered theory like Loomis and Sternberg.

If you want something high powered but not so much as that, i recommend Wendell Flemings book. And the book I mentioend above was erroneously attributed, it is by Williamson, Crowell, and Trotter.

heres one for 5 bucks.

Calculus of Vector Functions (ISBN: 013112367X)
Richard E. Williamson, Richard H. Crowell, Hale F. Trotter Bookseller: Frugal Media
(Austin, TX, U.S.A.) Price: US$4.75 [Convert Currency] Quantity: 1 Shipping within U.S.A.: US$ 3.00
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Book Description: Prentice Hall, 1968. Hardcover. Book Condition: Good. some stickers. Bookseller Inventory # mon0000109673

Last edited: Jan 31, 2007
7. Jan 31, 2007

### HallsofIvy

Staff Emeritus
One complication is that "Advanced Calculus" varies from college to college (and even from teacher to teacher) much more than the regular calculus sequence. In some places (typically engineering schools), it is literally more of what you saw in calculus with emphasis on things like Stoke's theorem, differential equations, partial differential equations, etc. In others (typically more liberal arts colleges) it is basically "mathematical analysis" with emphasis on the theory behind calculus. And, of course, all shades between. And there are "Advanced Calculus" books to match all of them.

8. Jan 31, 2007

### mathwonk

well, the first type is for many now epitomized by spivaks little book, calculus on manifolds, where he states clearly that the goal of the book is a clear presentation of the three main theorems of advanced calculus, namely fubini's theorem, the inverse function theorem, and the stokes theorem.

[oops he doesn't seem to say that at all clearly. ratyher he names three theorems: stokes, greens, divergence, as 3 basic theorems of classical analysis, but presents the other two i named. i still think im right about what message his books ends as to the three main thms of adv calc.]

i assumed it this sort of high level several variable calculus that was meant.

Is this the content of Folland? Or is it more of a first course in analysis, like baby rudin?

Last edited: Jan 31, 2007
9. Jan 31, 2007

### ayas

Everybody,
thank you for all your suggestions, I'll try to get a hand on as many as I can to be able to compare and find what seems good.
If you have other suggestions, don't hesitate to post and also please stress on what is good and/or bad about your book.

HallsofIvy,
to answer your question, this is an arts and science class, called 'Calculus of several variables', for math majors. And this is in a quite competitive and demanding University setting (University of Toronto precisely).

Fopc,
I skimmed through your suggestions, but at first it seems to be too fast-paced! I'd go for an undergrad book first.

Mathwonk,
thanks for the suggestions. However I already tried "marsden and tromba". I was happy and very satisfied at first, because of the intuitive approach and the numerous illustrations. But then, I also realised that this book is not as deep in its content as what I would need (see for example the section about limits which gives the general idea). That brings me to this question: is there a tradeoff? Is it either the intuitive approach or the deep content?
I believe not, and I also believe that math can be enjoyable and intuitive at any level... now I hope that a few bookwriters share that belief!
In your list of books of the 20th century, are there books that are valuable on both aspects?

Eccefeles,
Thanks for the idea but Folland goes probbly further than most 1st/2nd year books. For example, chapter 1 of Folland goes like this: Euclidean Spaces and Vectors, Subsets of Euclidean Space, Limits and Continuity, Sequences, Completeness, Compactness, Connectedness, Uniform Continuity. (the end of this 1st chapter is less likely to be really treated in first/2nd year calculus book)
Then the other chapters are:
2. Differential Calculus.
3. The Implicit Function Theorem and Its Applications.
4. Integral Calculus.
5. Line and Surface Integrals; Vector Analysis.
6. Infinite Series.
and so on

Last edited: Jan 31, 2007
10. Jan 31, 2007

### mathwonk

pretty fast paced. see if your library has the williamson crowell trotter book, or just order it for 5-10 bucks.

the old books like widder and buck are kind of old fashi\oned and maybe dry, but in the old days books were written with more detail and more motivation, since they appreciated better how hard the stuff is.

another book i like, but not too many illustrations, is Analysis I, by lang, if you can find it.

you need to look at a lot of them in a library and pick one you like. toronto will have dozens in their library.

folland sounds a bit like spivaks likttle calculus onamnifolds, superb but brief - you need to work every single exercise.

11. Jan 31, 2007

### Tom1992

i just ordered "analysis on manifolds" by munkres. nice theoretical approach to calculus in R^n. despite the title manifolds, it is only calculus of surfaces embedded in R^n, so it should be fine for the opening poster.

12. Feb 1, 2007

### fopc

Sadly (and for more reasons than you might imagine), my suggestion isn't suitable for this thread. Please accept my apology.

13. Feb 1, 2007

### mathwonk

well loomis sternberg is a classic, that is true, and he may benefit from it, I just thought he wanted more explanation. but you are right, if one can understand loomis and sternberg, then one has gotten a lot of good instruction. i took the course (math 55, harvard) from loomis in 1964-5 from which the book emerged, and although it was over my head at the time, still i did learn that a differential is a linear map, and that alone puts one ahead of the pack. he also gave a beautiful discussion of coordinate free affine geometry, not included in th book.

the book loomis used that year was the one by wendell fleming i mentioned above. i got a nice boost from that book too, on single variable calc, because there is an appendix summaizing one variable calc at the level needed for the book. it helped me a lot since i had bombed out of the honors level one variable cousre and was trying to recover. so by reading the 42 page appendix of fleming i picked up essentially all the point set topology i needed for the course.

then i also learned from fleming very clearly that the definition of a derivative in several variables is all about understanding the gradient of a real valued function of several variables. i.e. whether or not the target space has dimension greater than one is irrelevant (in finite dimensions).

i.e. the derivative of a function from R^n to R^m is just the vector of m derivatives of the coordinate functons. and he does a very clear job explainng the gradient.

one caveat if you are buying loomis sternberg, instead of getting it free, the binding of the reprint by jones and bartlett is apparently a cheap "perfect" binding with pages glued not sewn. i just noticed last night the front page of mine is loose. eventually they others will fall out. i love books and i feel cheated by this.

14. Feb 1, 2007

### mathwonk

the reason i recommended williamson crowell and trotter is it is very famous as one of the first books to try to introduce the modern lin ear algebra point of view, and they did a very careful job of explaining it, as often happens in an early version of something new. apostols books would also fall in this category. i.e. there is no volume two of spivaks book, but there is a volume two of apostols book.

loomis on the other hand apparently was trying to produce a work that would the "harvard" version of these 1960 texts. i can tell you from experience that did not include trying to make it easier, rather more high powered and elegant. but he is very clear. sternberg's writing and teaching is a different sort, his lectures were less organized and less beautifully written on the board, and so i avoided the class when he taught it.

later i learned by sitting in that he actually gave more insight in his explanations than loomis did. one wrote beautiful clear, but very formal lectures on the board. the other just turned around and told you straight up what was really going on. if you look at the book you may agree with me that loomis wrote the beginning probably chapters 1-7, and sternberg wrote the rest.

i liked the brief asides by sternberg such as appear on page 342, for why the change of variables formula is true. after taking loomis course, iw as blissfully unaware that although he had defined derivatives in infinite dimensions, he had not computed any. that gap is filled in the book partially by section 3.15, on tha calculus of variations, where eulers differential equation is seen to be the infinite dimensional equivalent of setting a derivative equal to zero to find a critical point. one needs this sort of thing even to show a straight line is the shortest path between two points! an example which is apparently not given.

15. Feb 1, 2007

### mathwonk

after all is said and done, you might try the two volumes of apostol, if they suit your taste. unlike spivaks "volume 2" which is small paperback of about 140 pages, apostol continued in vol2 as in vol 1, with a complete treatment of several variables.

but you might compare also crowell et al. and ewvewn the older books above. i do not know just what it is you need review of, and if you are like me, different books will help with different topics.

e.g. although loomis course did not entirely suit me, he made many things crystal clear, partly by his clean notation, e.g. the proof of the chain rule.

to wit: a diffble function has form L+o where "little o" is a function whose graph is tangent to the source axis and the linear function L is the derivative. thus the composite of two of them looks like (L+o)(M+o)

= LM + Lo + o(M+0). now the composition of any reasonable function with a little o function is also little o, and the composition of two linear functions is linear. hence our composition looks like LM + o, so the derivative of the composition is LM, the composition of the derivatives.

hows that for simple? see thm 6.2, chapter 3, loomis sternberg pages 143-4.

and although the proof is clear, you will see not one word pointing out that it should be obvious that the best linear approximation to a composition is the composition of their linear approximations.

Last edited: Feb 1, 2007
16. Feb 1, 2007

### mathwonk

one final remark: another great high level advanced calculus book and the one that started the trend in the 60's is the famous "foundations of modern analysis" by jean dieudonne. unfortunately, dieudonne was apparently kind of a madman who eschewed all diagrams and illustrations, and there is not a single one in his book, a point he makes in the introduction with some pride.

it is also somewhat hard to find at any reasonable price, but libraries will have it.

17. Feb 1, 2007

### mathwonk

will wonders never cease, suddenly dieudonne is out there for sale:

this is the time for those interested few to buy it:

Foundations of Modern Analysis
Dieudonne, J
Price: US$13.00 Book Description: Asian Imprint, 1960. Book Condition: Good to Very Good. Hardbound. oops! its a pirated ripoff copy!! i refuse to participate in this thievery. but heres the real thing (apparently) for only a few dollars more, as they say! Foundations of Modern Analysis J, Dieudonne Bookseller: Bertram Books (Boulder, CO, U.S.A.) Price: US$ 14.95
[Convert Currency]
Quantity: 1 Shipping within U.S.A.:
US$3.00 [Rates & Speeds] Book Description: Academic Press. Book Condition: Very Good. 1960; Hardcover, no dust jacket; Good copy has clean text with no marks; ownership marks on front endpaper; binding firm; cloth covers with minor spotting and wear, else intact. Bookseller Inventory # 115520 Last edited: Feb 1, 2007 18. Feb 1, 2007 ### mathwonk it sems the pirated editions ahve forced down the price. the unpirated later volumes of treatise on analysis are still over$100. as well they deserve to be in a world where a crummy calculus book like hass weir and thomas is that much.

Last edited: Feb 1, 2007
19. Feb 1, 2007

### Tom1992

wow! i never learned the derivative like this before. let's compare:

university calculus: f:R^m -> R^n is differentiable at a if there exists a nxm matrix T such that

lim(h->0) |f(a+h)-f(a)-Th|/|h| = 0 (boring)

loomis and sternberg calculus: f:A -> W is differentiable at a, where A and W are Banach spaces, if there exists a linear map T from
Hom(A,W) such that

f(a+h)-f(a) = T(h) + o(h) (wow!)

and look at this: f ' :A-> L(A;W) (linear map), and f '' belongs to C^0(A;L(A;L(A;W))). furthermore, the chain rule for (h o g)'' is given by:

(h o g)''(x) = h''(g(x)) o (g'(x),g'(x)) + h'(g(x)) o g''(x)

nice, abstract and beautiful! none of that limit rubbish. i got to read this book!

Last edited: Feb 1, 2007
20. Feb 1, 2007

### morphism

I would say they are equally "boring" because they are the same thing.

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