Customer Reviews

Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (Second Edition) (Volume 1)

5 star:		(26)
4 star:		(2)
3 star:		(3)
2 star:		(0)
1 star:		(2)

 

 

Apostol's presentation differs from the standard order and content for a calculus course, but is the more useful for it. Introducing integration first is historically more accurate and sets the tone for the rest of the book. This is not a "plumbers" book but the examples inform the abstraction very well. This book does not bog down in the tedium of analytical geometry and figure recognition which is too often the case elsewhere.

I am using the book for self-study as a middle-aged adult and find the presentation makes sense of things from other sources. The intellectual level is demanding but not unreasonable--challenging without being overwelming. While the introduction of linear algebra may no longer be needed for introductory calculus students, presenting it in the context of the calculus ties thing together nicely.

This book is extremely well-written and leaves you with the feeling that it couldn't have been better. A tribute to this fact is that it is still in its second edition from 66 and, though it is rather old, has kept its quality.

It has a good number of exercises (usually between 15-30 per section/topic), which is less than most standard calculus book, but the difference is that the quality of the exercises here is much higher, and you will be surprised when some months later, when tackling some problem for another course, you will remember having done the exercise in Apostol. It also has answers to all the exercises (except for the ones which require a proof, rather than a number as a result). The problems range from easy to very hard, but usually there won't be more than two problems per section that one won't be able to do upon first reading and a little thinking.

The writing of the book is very good and rigorous, and it covers some topics that are not present in most calculus books. For example it has a small seciton on partial derivatives, it covers the weighted mean-value theorem for integrals and rearrangements of series. There are many other topics that don't usually fit in a calculus course, but the introduction of these when you are still learning it makes the connection between the topics much clearer. After having read the book from cover to cover, it has now become a very useful reference that never leaves my table. Also, because it is rigorous and has a broad number of topics, if you learn this and vol. II now you will save a lot of time later in more advanced courses such as analysis, differential equations, linear algebra and to a lesser extent even differential geometry and probability.

Because of its nonstandard approach, I think that this book is unsuitable for most people learning calculus for the first time (especially if you are taking a course and not just studying at your own pace). However, it (along with vol. II) is mandatory reading for anyone who wants to study math, in my opinion.

Perhaps the best description for Calculus, by Tom M. Apostol, is simply its title. This text is Calculus. Like no other calculus book I have seen, it devotes itself totally to its subject matter, never compromising itself for the sake of understanding. By doing this, the reader is permitted to learn calculus completely.

So many calculus texts in the current market have a sort of misguided focus. Instead of explaining the subject they claim, all they offer is the tools for solving the rote calculus problems of Advanced Placement tests and engineering. This is fine for someone who cares nothing of mathematics, but is not sufficient for their claim of teaching calculus. Apostol's Calculus cares little about explaining the applications of calculus or preparing someone for yet another standardized test. Uncluttered by fancy computer-aided graphics and pages and pages of redundant examples, Apostol offers the basics of calculus with the prrofs behind the theorums. Never once is the reader left with questions as to what exactly integrals are or why any two nonequal numbers must have another number between them. Everything necessary for the reader to solve any single variable calculus problem is presented in text. Apostol's rigor knows no bounds, begining first with the proof of the positive integers and continuing to the finest points of integral calculus.

This text is not for the faint-hearted. If you just want to be able to solve calculus problems, you would have little use for this text. But if you want the tools and justifications for all of calculus, this is the book for you. It is a necessity for all mathmaticians' libraries.

See also Calculus 2 by Tom M. Apostle for multivariable calculus.

This book, unlike most Calculus books out there, is meant to be read and understood. The way that Calculus is taught nowadays, people use books that are 90% problems, exercises, and examples, with an emphasis on computation. This book is 90% prose, and the emphasis is on cultivating a deep understanding. In addition, the book does away with the gap between "Calculus" and "Analysis", choosing to begin with a more mathematically mature perspective...but providing ample explanation for students who have not seen the material before.

This book is exceptional for self-study. I would recommend it to anyone learning calculus on their own, who actually wishes to understand it. This would make an excellent supplement to one of the standard Calculus textbooks, since it addresses just about all the classic weaknesses of these texts. I wish colleges would use this as a textbook, but alas, that would require a drastic restructuring of the curriculum.

This book may come across as "hard" to students, but this is only because it is structured in such a way that one cannot not get through it without understanding the material. Also, a student finishing this book will be ready to dive into more advanced analysis courses, whereas students using basic intro calculus textbooks will find themselves very poorly prepared for these things. The current calculus books with their emphasis on mechanical computation, allow students to get through without understanding the material, and that is why they come across as "clearer". In reality, they are much less clear than this book.

Frankly, it is too bad that modern academic institutions and those responsible for it's direction have forgone the use of two marvelous and impeccably well-written Calculus texts and in their place have opted to baby students with such authors as Stewart. In the hands of a confident and versed instructor these two texts are worth their weight in gold since they carry the student through the normal Calculus I, II and III sequence as well as providing a nice digression into Linear Algebra. As stated by prior reviewers, these two books are complete, rigorous, Apostol never cuts corners in his presentation of the material and he shows the student exactly how calculus and in general mathematics texts should be written. By far these are best and everything else is merely a waste of paper. My hat goes off to Apostol for continuing to his legacy of well-written Mathematical texts

if you are the kind of student who thinks the area of a triangle is: "Mmmm...(1/2)BH, or is it (1/3)BH? Oh well, who cares, I can always look it up anyway." then this book is probably not for you. But if you have wondered whether the number given by that formula changes when you change which of the three sides of the triangle you call the base, then this book is exactly what you have been looking for. The author of this book assumes you are bright and curious, the kind of student it was written for at Cal Tech. Professor Apostol has made an enormous and thoroughly successful effort to explain in intellectually honest detail exactly what is going on, and how everything is proven rigorously. In this same vein, Spivak's book is more fun, and Courant's has more physics and applications, but this one is the most scholarly. The linear algebra was an afterthought, added to the original work when that craze swept the subject in the 1960's. In general, great works are best in the first edition before the publisher convinces the author to modify his original vision. In my opinion the linear algebra does nothing to enhance the presentation of one variable calculus, and should have been left out, but it won't hurt you.

This is one of the best math books I have ever used; Apostol does a great job of giving you intuition and telling you why the topics are important. The exposition is very clear.

However, this is not your usual Calculus I book. The level of approach is halfway between an intro to calc class where you learn HOW to calculate derivatives/integrals, and a real analysis class where you learn why derivatives and integrals are related and their axiomatic foundations.

Bottom line: this is a great book if you want to understand calculus deeply. It is a warmup for real analysis. However, there are probably better books if your only interest in calculus is for using it to solve engineering or physics problems.

This book was used for my introductory calculus class. Though some reviewers believe that this text isn't suitable as a first introduction to calculus, I didn't have any problems with it, and as far as I know no one in my class did either. Quite the opposite, in fact. The detailed proofs and derivations made material that might otherwise be difficult easy to understand. The introduction of linear algebra before multi-variable calculus (which is covered in volume II) makes some of the more abstract concepts in multi-variable calculus quite straightforward. In general, the proofs are detailed enough for someone with no prior experience in a proof oriented class to understand, but concise enough to not be tedious

Four years after having taken my first calculus class, I still remember how to apply techniques such as taylor series and lagrange multipliers because, rather than just throw out formulas and laundry lists of steps, Apostol goes through derivaions that engender a deep understanding of the mathematics behind the techniques.

The name above should read: Old-Time Math HOBBYIST!

Sorry about that! Oh, the vagaries of aging ... to the review of Apostol One:

Oh, the memories!

Young mathematicians, kindly grant an old-timer the license of your tolerance. I thank you in advance for your patient forebearance. Sigh, ... the memories ... good ones, to be sure:

The time ... Autumn 1973. The place ... The University of Chicago. The event ... Yours truly in "Mathematicians' Calculus Class" ... ! Back then, they did not call it Honors Calculus, as I believe they do today. Also, it was The U of C, rather than UChicago, as it is now.

Then, as now, you arrived at The U of C and spent your first week of orientation taking placement tests. By some unbelievable miracle, I placed into The U of C's very tippy-top math class for entering freshman. Most of my classmates had a perfect 800 SAT math score. The few that did not have the perfect 800 had like a 790, 780, 770. Hey, close enough, right? Me? Not in that league. I got by on determination alone and nothing else. There were only two sections of this class, and each section had no more than 15 people in it. The point is that only the best of the best of the best even qualified to sit in that class in the first place.

Back then, they used Apostol for that class. Nowadays, I believe, they use Spivak; however, I'm not entirely sure about that.

Anyway:

This book was then, and remains today, a veritable GEM for anybody truly and intensely interested in mathematics for its own sake and not merely as a tool to use for some profession or another. That's why they (The U of C Mathematics Faculty) called our class "Mathematicians' Calculus Class," and, again, Apostol One was the textbook they chose for that class.

Tom Apostol of Caltech wrote the definitive introductory calculus book FOR MATHEMATICIANS back in 1966. When I studied from it, it had only been out for 6-7 years or so and was only used at no more than six mathematics departments in the entire country. I believe they used Apostol Volume One (We didn't know it as Tommy 1 a way back when) at Caltech (of course!), Princeton, The U of Chicago, I believe at Michigan. I'm not sure of the other place or two. It's been nearly 40 years.

The Chairman of the U of C Mathematics Department back then chose this book, because he and the faculty truly believed that:

This is THE CALCULUS BOOK for anyone aspiring to become a mathematician and, indeed, a pure rather than an applied mathematician.

The writing is clear and concise. The treatment is rigorous. The emphasis is on proofs. The problems range from a little challenging to hard as hell.

Those of you with a true gift for mathematics (If you are using this text you have been deemed to have that gift, or you would not be using it for your class, I assure you!), will relish reading this fine classic in mathematical didactics.

Notice, I only gave Apostol One four stars and listed a reservation in the title to this review. Here's why:

This book is TECHNICALLY BRILLIANT ... BUT !!! ... it is rather a bit dry, as most mathematics texts seem to be. (Parenthetically-speaking, IMHO, the driest and most terse of all such books is Walter Rudin's "Analysis" [The shortened title!])!

What, indeed, does Apostol One lack?

THE PASSION, PROSE AND PURITY of teaching offered to us by Professor Spivak! Spivak veritably scintillates with excitement! That empassions its readers! THIS is how to teach mathematics ... and any other subject for that matter. The purists will not approve of Spivak's delivery and will prefer Apostol's. Hey, live and let live, I say. I'm cool with that. To each their own. Personally, I genuinely believe that Spivak's passionate (albeit literary) approach to teaching is the way to go.

The ultimate calculus book intended for aspiring mathematicians would be, IMO, an admixture of both Apostol One and Spivak that blends the best features of both tomes into one supremely marvelous course text!

Said book, to my knowledge, does not yet exist. The solution? Why, read BOTH Apostol One AND Spivak, of course!

Lastly, our sophomore-year text for "Mathematicians'-Track Mathematics" at The U of C back in The Day was, you guessed it, Rudin's "Analysis." Yawn ... !

All in all, Professor Tom Apostol's Calculus, Volume One, is truly a masterpiece. Buy it. Read it. Love it.

Be aware of one caveat, however: This book is aimed at a mathematically-sophisticated audience; i.e., students who have either genius-level intellects or lots of mathematics experience.

The determined non-genius can read this book and genuinely benefit from it by dint of determination. I am that kind of person myself. If I can enjoy it, many of you can too ... provided that you have either the genius or the determination. I got by on the latter. You can too: "Ya gotta wanna!" If you do, you're in for a real treat with Apostol One.

Thanks again, young mathematicians, for putting up with the ramblings of this old-timer. Down the line, when I finally do retire, I'm going to study the bejeeziz out of these wonderful mathematics books just for the ever-lovin' hell of it ... and have a stone-cold blast doing so! Sooner than you think, you, too, will be along in years and have very fond memories of things like this. Trust me, you really will. You'll see.

Happy Mathematics for many years to come. My best to all of you. Read and enjoy this book. Tell a friend about it too. I salute you!

U of C Old-Timer

Apostol's Calculus, Vols 1 and 2, are my calc bibles. It is unfortunate that more undergraduate curricula and textbooks don't follow the axiomatic, yet cleanly-written style of Apostol. Standard texts (e.g. Stewart) pull theorems out of the error without proof, as if the point of taking calc is to get it done and over with as quickly as possible.

To be sure, Apostol isn't for everyone. If you are fascinated by color pictures and your primary interest is to learn the bare minimum, buy another book. But if you want to get more out of your undergraduate calc, whether you're a math or science major, or you're studying on your own (I meet both criteria), Apostol is the way to go.

This is a must if you're thinking of taking any higher math. Worth the bucks, it'll save you in the long run.