Customer Reviews

Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability (Volume 2)

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3 star:		(1)
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Apostol's Calculus is the definitive book on Calculus for anyone who wants to be a mathematician. Historical notes, intuitive ideas, clear definitions, demonstrations, all is there, from natural numbers to Stokes' Theorem. His applications of linear algebra to multivariate calculus are among the best I have seen on calculus textbooks Better than this, only a book on Mathematical Analysis.

Few books in the mathematical literature have given me so much pain as this one. Freshman year, I took a heavily theoretical linear algebra class with Tommy II as the textbook, and then the next term I took multivariable calculus out of this book as well. In either case, this book was my first experience with the material, though as an "introductory" text it should have done the job. Suffice it to say that neither experience was terribly positive.

My problem is that Apostol never seems to try to motivate ideas well, and he uses cumbersome, nonstandard, and occasionally inconsistent notation. His proofs can be inelegant and opaque at times. He is far too sparing on geometrical intuition as a way to understand the material, preferring to talk in symbols rather than pictures. (This is especially true in the first five chapters on linear algebra. His multivariable chapters are well-illustrated, but calculus on R^n seems to be trivial once calculus on R is under your belt from a good introductory book like Larson/Hostetler/Edwards at a high-school pace. Thus, the motivation is needed least where it is used most.) As a result, I feel that I still don't intuitively understand how operators work on inner-product spaces, even after trying to remedy my deficiencies for a year and a half now.

I attributed my lack of understanding to my stupidity, but then I found myself learning exterior forms from Arnol'd's excellent mathematical mechanics book and groups from Dummit/Foote's superb abstract algebra text - and understanding the exposition perfectly. And I started to feel that this book is the thing at fault.

If a prospective reader is prepared for the terseness and difficulty of Apostol, I recommend that s/he go straight to the real math rather than settling for this obfuscated treatment of inroductory subjects. It is no harder to learn the rudiments of metric topology than it is to learn Apostol's open balls, and it seems no more inspired to take on Halmos' linear algebra classic, with its intimations of Hilbert space, than it is to struggle through Apostol's treatment. (The former seems to combine considerable difficulty with terse, but wonderful, motivation, but don't take my work on that: I'm only forty pages into it!) But the books are more inspired, and the math is far more general and beautiful.

My recommendation: learn your calculus (and potentially your first linear algebra) patiently but thoroughly from a prosaic, worked-example-ridden, 1000-page monster, then go straight to the upper undergraduate/early graduate classics for the real fun. Tommy II, caught somewhere in the middle, has no place in this plan.

I'm currently taking an honors calculus sequence at the U of WI, and have used this book and the first volume for the past three semesters. Needless to say, you have to take Apostol with a grain of salt. Although the no-frills style and lack of worked examples is upsetting to many students who are used to pictures, thorough examples, and color, these volumes cover a lot of material in a small space. And also beware; my professor and others in the math department have found errors in definitions and theorems, and the archaic notation is off-setting at times. Basically, if you're looking for straighforward information (written by a mathematician, for a mathematician), you've found the perfect book. If you're looking for an easy-to-read and understand book, keep searching.

I was looking for a solid reference book and was quite fortunate to stumble across Apostol's two texts. His writing is clear and concise. What I appreciate most is his axiomatic approach. He builds up everything as opposed to the numerous calculus cookbooks out there. Every theorem has a proof.

The author is a world renowned mathematician, with many beautiful achievements under his belt. I would not include this popular textbook in this category.

I tried to use it for my real analysis class but I decided otherwise when I opened the first volume. I believe strongly that the key concept underlying real analysis is the concept of convergence/continuity. What turned me off from this book was his consistent effort to avoid this concept for the first half of the first volume which deals with rather sophisticated convergence problems involving Riemann/Darboux sums.


The choice of opening a beginner's book on analysis with Riemann integral is rather unusual. In itself, being unusual is not a negative, but in this special case I do not believe it is helpful. I believe that in learning one should start with simpler examples and gradually increase the difficulty. In this book the reader is thrown in some of the most complex situations, while deliberately avoiding the two ton gorilla in the room. This is not how this reviewer and many of his students learn a new theory.

What is then a young reader to do? I have two strong recommendations. The first is the classic text by G.H. Hardy, A course in pure mathematics. This old classic is still relevant today. Hardy was an elegant writer with a beautiful mind, arguably one the best analysts of the 20th century. His book has rigor, geometric intuition, beautiful examples, and a genuine empathy for the green mind. Some exercises in this book can be a bit challenging, but always very rewarding.

The second recommendation is Terry Tao's textbook on real analysis. His is arguably one the the best analysts alive, and he has a keen sense of the traps awaiting the modern students. As far as organization, it follows the standard structure, which I believe was a wise choice. He is not out to blow your mind with his expertise, but to slowly and carefully peel off the layers of mystery.

One more argument in favor of Hardy or Tao's books: the price. For any of them, you will pay roughly half the price of Apostol's book, and you will get more in return.

I am currently enrolled in BC Calculus in my high school as well as linear algebra at a local college. What better way to learn both together than with Tommy. This is a great book to learn the connections between the two and how to do real linear algebra, not straight algebra but differentiating and doing calculus on whatever spaces you want. It's very concise, however not so clear. I skipped into BC and spend a lot of free time doing math and this book is still a bit deep. Also, the tie-ins to LA are definitely not going to be apparent off the bat. I have a really great LA teacher so I find myself skipping over some of his more complicated expressions of very simple items, however if i were a newcomer to LA, this would be totally confusing and Greek. I agree with the other reviewers, if you're familiar with calculus and LA and want to learn more about each and their connections, this is the bible, however, if you're a newcomer to one or both, definitely learn each separately and more simply. The book is very proof based and states it assumes you know how to use the mathematical objects it's presenting, now it's showing you why they work. Some of his expressions are like physics problems mindset, first look you'll have no idea, but if you think about it, eventually the ideas all fall together. A great book and recomended to anyone experienced enough to handle it.

if you have gone through the vol 1, then you do not need my review. I feel vol 1 and 2 are the best calc books for math majors. There is just enough theory to make the transition to analysis very smooth. These 2 books cover calc 1-2-3 and diff eq. plus some more. The author is very clear. Look for a cheaper copy on-line.

This volume is the hammer of the two volumes, where as the first volume would be the nail. Once you've master volume one it's time to start digging for gold; the basics one learns in volume one will be expanded to multi-variables and applications that will challenge even the most skilled students. The introduction of linear algebra is a good preview of the things to come for those interested in pursuing mathematics even further.

This book contains a lot of information, and is rigorous, with many proofs and a vast array of problems. However, it is weak on worked examples and in explaining the concepts clearly. Diagrams are kept to a minimum. I used this book during an undergraduate Calculus course, and found that it did not help me to grasp the fundamentals of the subject. There are much better books available e.g. Stewart, which cover much of the same ground in a much simpler format. In my experience, only the ablest students were able to benefit from Apostol's dense writing style and scarcity of worked examples. Apostol does, however, include sections on matrix algebra and calculus which are not available in many other textbooks. There are also solutions to many of the problems at the end of the text.

I used these books (Vols. 1&2) in my last two years of high school back in South America. I remember long nights of bad coffee and cigarettes locked in my room reading it over and over. The book is full of really cool knowledge (even for a non-mathematician). Most of today's students learn like little parrots, (without thinking or understanding) just repeating things mindlessly. This will make this book unpopular among these people because having to read a sentence and stop to figure things out on their own is too hard a challenge for them. I guess that like the high standard of the education of old when teachers loved teaching and their subject this book is also going the way of the dinosaurs. I'll get my copy before that if I were you though ;-)