Spivak as Preperation for Mathematics Degree

jgg

Hi,

I got a 5 on the AP Calculus (BC?) exam, so I have a basic knowledge of calculus (probably on the level that a Stewart book would teach). However, I'm planning to major in mathematics and computer engineering/science, so I'm looking for a rigorous introduction to Calculus. I've read the first chapter of both Apostol's book and Spivak's book, and I really like both. However, I like Spivak's problems, since they really force you to think about the material. However, I was concerned when I found a few posters (I think it was on reddit) who said that Spivak's text is 'incomplete' for teaching Calculus. If I wanted to go to grad school for math, is this the case (i.e., how far does Spivak's Calculus go?) or are people just being stupid on the internet (again)?

snipez90

huh, I probably wouldn't take everything people say on reddit seriously. spivak is fairly complete, but I feel like there is so much calculus you can learn before moving on. some of the problems in the book are taken from analysis texts, including rudin's PMA.

jgg

This was a complaint I found on more places than reddit, but, yes, I know better than to believe everything coming from that bunch.

Could you please clarify what you mean by this?

I believe this was the book where he suggested in the preface that it should have been labeled as an analysis book, correct?

snipez90

I think you may be referring to this thread:

http://www.reddit.com/r/math/comment...oks_would_you/

So hmm, it seems there is a fuss over the lack of geometric intuition and applications. Well firstly, the text does have a lack of applications because that's simply not the focus. The solution is simple: find a textbook with applications if you're interested in them. But if you want to understand the theoretical underpinnings of everything you did in Calc BC, this is the book for you.

As for the geometry, I don't think Spivak's calc text moves from presenting concepts very abstractly early and then presenting them geometrically later on. He maintains a fairly good balance throughout. Besides, developing geometric intuition is partly the reader's job anyways and is perhaps the most useful tool in solving many of Spivak's problems.

*EDIT* With regards to the preface, yes I think so. Although you'll be familiar with much of the terminology used in Calculus, you'll see by working through problems that the experience will likely be much different from AP Calc

jgg

That's the one. I found similar complaints elsewhere. Th

Theoretical is what I wanted. I wasn't expecting applications. If I wanted those, if I remember correctly Courant is nice for that.

I've already started looking at exercises in the book, and I enjoy doing them. It's less of the 'plug and chug' crap that is perpetuated throughout the AP classes.


I guess what I was getting at in the first post is this: How much Calculus does this book actually cover? Do I really need to buy all of his books, similar to Apostol, to get the full picture?

snipez90

Yes, but Courant has a theoretical leaning as well, but I don't know how it compares to Spivak since I have never actually read Courant.

Oh sorry, I usually take "calculus" to mean topics in elementary calculus presented at varying levels of rigor. Spivak focuses on functions of a single variable, and many concepts of analysis are introduced along the way (and many more if you look through the exercises). Thus, you won't find any multivariable calculus in here. Spivak has a small but decent intro to complex variables after discussing sequences and series (of numbers and functions), but besides that, he sticks to real functions of a single variable.

This shouldn't be that discomforting though. Calculus on Manifolds I think is used in Analysis in R^n courses. Differential geometry is also a more advanced subject that might be found in a course after analysis.

jgg

Ah, exactly what I was asking.

Thanks a lot!

jgg

So, would it be more fluid to go between Apostol I & II to do a complete study of Calculus?

EDIT: Obviously, no study is 'complete', but you get the idea.

snipez90

Eh, Spivak covers pretty much the same topics as Apostol I, I think. I've read a bit of Apostol and I personally found Spivak easier to digest. Although Apostol may seem a bit dry compared to Spivak, Apostol is good at explaining the material. Either Apostol I or Spivak will prepare you to tackle Apostol II. Also, the linear algebra introduction towards in the end of Apostol I is in the beginning of Apostol II, I think.

jgg

Spivak is much cheaper than Apostol I, so maybe I'll do that. Thanks for your help.

JG89

I've read most of Courant, and a bit more than half of Apostol.

I personally enjoyed Courant more. I thought he gave more intuitive proofs and I also like the tone in which he writes. I believe Courant is on the same level as Apostol and Spivak (I've read a bit of Spivak) in terms of rigor.

For what it's worth, Courant contains more material than both Apostol and Spivak, with Apostol also containing more material than Spivak.

jgg

JG89 what is your level of mathematical maturity (if you don't mind me asking that, I figure it's relevant)?

EDIT: and what is the title of the courant book you're reading? Differential and Integral Calculus I? I seem to recall that there's another one with his name on it floating around...

JG89

The book I have is Introduction to Calculus and Analysis

Courant was my first exposure to theoretical calculus. I read about 75 pages of it before I realized I didn't learn anything at all because I simply didn't understand the material.

I dropped Courant and started reading Apostol. I understood the material in Apostol and that developed my mathematical maturity a bit. Then I took a glance at Courant one night and I was understanding the material, plus I enjoyed the writing more (Courant is more conversational in tone), so I dropped Apostol and started Courant again from scratch and haven't looked back since.

JG89

Here's an online copy of Courant's differential and integral calculus:

http://kr.cs.ait.ac.th/~radok/math/mat6/startdiall.htm


It differs a bit from his introduction to analysis, but for the most part it covers the same material.

jgg

Hmmm...I think this link is to his "Differential and Integral Calculus." If I'm reading correctly, the one you have is a rewrite, which is newer and therefore more expensive. What do you think of Spivak? I like Courant but I'm hesitant to drop Spivak...

Howers

If you're getting Courant get Calculus and Analysis, with that other author - its the new edition. I think Courant is better reading, and is more true to calculus. Spivak is good because of his insane problems. And because problem skills, not reading, define a mathematician it is the best book. Spivak's reading is also very good, but it is a little disorganized and cluttered. Its not a biggie when learning, but when you refer to it after a few years it is annoying.

My suggestion to you however is to strengthen your basic mathematics. I made this thread for that purpose: http://www.physicsforums.com/showthread.php?t=307797
In particular, if you don't know how to prove this thread will be more useful to you than Spivak.