- Hardcover: 680 pages
- Publisher: Publish or Perish; fourth edition (July 9, 2008)
- Language: English
- ISBN-10: 0914098918
- ISBN-13: 978-0914098911
- Product Dimensions: 10 x 9.2 x 1.5 inches
- Shipping Weight: 3.8 pounds (View shipping rates and policies)
- Average Customer Review: 123 customer reviews
- Amazon Best Sellers Rank: #133,620 in Books (See Top 100 in Books)
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Calculus, 4th edition fourth Edition
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Furthermore, I do not believe it is possible to truly understand what a limit is if you don't read something like this. I am appalled at calculus curricula at colleges and universities that just expect you to memorize rules and drill problems over and over; I prefer knowing exactly why something is true before doing any problems. I also prefer fully 100% understanding every single reason a theory works, so this book is perfect for me, though it doesn't construct the real numbers from the rationals. This is a prerequisite to understanding limits because if you don't understand what irrational numbers are first, then you'll never understand what is meant by "infinitely close," period. So I recommend you buy "Understanding Analysis" by Stephen Abbott with this book, and read ch1 of Abbott before this. That way you'll know what a real number actually is, by the way, it's the limit of a sequence of rational numbers, but don't worry, Abbott ch 1 is incredibly well written and you'll get it if you just read it and think about why it makes sense. If you want a little deeper of an understanding than Abbott (which is splitting hairs, and few could pull it off, but the man I'm about to name is one of the few who can) then take a look at "Analysis 1" by Terrance Tao. I'd read either of these analysis books before or concurrently with Spivak, and I might also suggest grabbing a copy of "Single Variable Calculus: Concepts and Contexts" 4th edition, by James Stewart in order to see the applications of things like the integral and in order to get a better idea of how modern courses have a working understanding of the subject in general. Sometimes it's just good to be up to date with things.
Spivak manages to deliver both an intuitive picture of a concept and the full mathematical rigor in a brilliant and playful style. He will often give a provisional definition of a tough concept to aid understanding first, but importantly and in contrast to more "accessible" math books, he signals very clearly that he is being intentionally imprecise. He then moves towards rigor by explaining exactly the way in which he has been imprecise, clearly driving the motivation for a more rigorous definition. The overall effect is that you rarely feel very lost and when he ultimately gives you the full picture, it often feels like an inevitability. A favorite example of this sort of style is at the start of Chapter 20: "The irrationality of e was so easy to prove that in this optional chapter we will attempt a more difficult feat, and prove that the number e is not merely irrational, but actually much worse. Just how a number might be even worse than irrational is suggested by a slight rewording of definitions..."
Another impressive aspect of the book is the layout, where every relevant figure is only a glance away in the margin or directly inline with the text. It is the same style used in the Feynman lectures and Edward Tufte's books, and it is executed at its highest level here. Clear care went into the placement of each symbol in each equation and each figure.
The exercises are quite hard, but there is a full solutions manual available for self-study (how I am working through the book). I will admit that I needed to bail out of this book at the very beginning, never having been exposed to doing proofs at this level before (formulaic high school geometry "proofs" don't count for much here). I used Velleman's "How To Prove It" and the first few chapters of Apostol's Calculus Volume I to get up to speed. Both these books are also recommended, and Apostol, in particular, gives an excellent and rigorous but more gentle on-ramp for the sort of thinking asked of you in Spivak Part I. In the long run, however, I think Spivak edges out Apostol for self-study because of the solutions manual.
I picked up this book when I found that after 3 years of doing calculus in high school and college, I had forgotten most of it within a few years. I realized that while I could do the mechanics, I never really understood calculus in the first place. This book is probably a bit of overkill for just patching understanding, but I now have a much deeper appreciation and understanding of the mathematical way of thinking. It's not an easy book, but it is a wonderful one that will pay back dividends for hard work.
But you don't have to do all the hard work just to appreciate what Spivak has done here. If you have an interest in good writing, this book is worth a look even if you aren't interested in learning the subject. I take special pleasure in reading great writing on any topic, and this book is up there with the best writing anywhere.
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The material is written with extreme clarity and the organization of the material...Read more