Is Spivak's "Calculus" Enough for an Analysis Course?

[mitcho]

Is Spivak's "Calculus" rigourous and comprehensive enough for an analysis course?

I will be doing my first course in real analysis this year and the textbooks have just been released. The recommended text is Spivak's "Calculus" as the title suggests. I have a copy of Stewart's "Calculus" and have heard that the content is very similar except that Spivak is slightly more rigorous. I am a little skeptical that Spivak (based on what I have heard) will be enough for an entire analysis course and so was thinking about getting a specialized analysis text. Would you recommend this or not? Do you think Spivak will get me through?
Thanks

 

[QuarkCharmer]

I'm not sure if it's okay for an Analysis course (having not taken one myself), but I can tell you that Spivak's Calculus is NOTHING like Stewart. Not even remotely similar, it's so much more difficult you have no idea. Try to look at some of the problems of a section you are familiar with in amazon preview or google books or something...

 

[mitcho]

That may actually be the answer I am looking for... I heard it was slightly different but I thought that was only in the approach. If it is much more difficult as you suggest then it might just be rigorous/theoretical enough for analysis perhaps?

 

[micromass]

As an ANALYSIS course?? No, Spivak is not good enough for an analysis course. And Stewart is certainly not good enough, it's not even good enough for a calculus course in my opinion.

Spivak is in a way a bridge between calculus and analysis. It's a little too rigorous for calculus, but a little too easy for an analysis course.

The problem with Spivak is that it simply doesn't treat the right topics for an analysis course. It treats the normal calculus topics. But it doesn't get into the fun analysis things.

Then again, if your recommended book is Spivak, then just go along with it. But it won't be the rigorous analysis we're all used to.

Check my blog for a collection of analysis books.

 

[jbunniii]

Spivak's treatment of calculus is closer in spirit to an analysis course than most calculus textbooks. If you learn from Spivak, you will have an easy transition to analysis texts such as Rudin's because you will already be very familiar with rigorous epislon-delta limiting arguments: why they are necessary, how to recognize when they are correctly done, and how to write your own. You will understand calculus at a more sophisticated level than if you learned it from Stewart. You will probably appreciate and admire Rudin's slickness rather than being bewildered by it.

Spivak doesn't cover any topology, and as I recall most of his theorems are for closed bounded intervals in R^1. An analysis course will generalize a lot of this material to arbitrary metric spaces, with R^n as a special case. This generalization actually makes a lot of the proofs cleaner, in addition to extending their domain of applicability. Compare Spivak's "Three Hard Theorems" with the analogous material in Rudin to see what I mean.

 

[JG89]

It might help if you post a quick summary of topics your course will cover.

 

[mathwonk]

In the old days, the progression was roughly: rigorous one variable (Spivak) calculus, Abstract algebra (Birkhoff and Maclane), rigorous advanced calculus (Loomis and Sternberg), introductory real and complex analysis via metric spaces as in Mackey's complex analysis book, general analysis as in Royden, (big) Rudin, or Halmos and Ahlfors, algebra as in Lang, and algebraic topology as in Spanier. Then you specialize.

In particular Spivak was written for a first semester freshman book.

 

[mitcho]

It might help if you post a quick summary of topics your course will cover.

This is the concise course summary and I have to say it doesn't look nearly as rigorous as analysis courses I have seen at other universities:
"Bounded & monotone sequences. Sequences & series of real functions. Intermediate & mean value theorems, iterative procedures. Taylor's Theorem & error estimates. Criteria for integrability. Vector functions, continuity & differentials. Implicit & Inverse Function Theorems & applications. Multiple integrals."

In particular Spivak was written for a first semester freshman book.

This was my concern, Stewart has gotten me through Calc I, II, III and so I am not sure if Spivak would be much of a step forward?

 

[JG89]

This is the concise course summary and I have to say it doesn't look nearly as rigorous as analysis courses I have seen at other universities:
"Bounded & monotone sequences. Sequences & series of real functions. Intermediate & mean value theorems, iterative procedures. Taylor's Theorem & error estimates. Criteria for integrability. Vector functions, continuity & differentials. Implicit & Inverse Function Theorems & applications. Multiple integrals."

Spivak will cover up to criteria for integrability. He will cover it very thoroughly and rigorously. You will need another book for the rest. You might want to try introduction to calculus & analysis volume 2 (which deals with multivariable calculus). His volume 1 deals with single variable calculus in a rigorous fashion, similar to Spivaks. So you may even want to check out both volumes.

 

[mathwonk]

more precisely, spivak was written for a first semester elite honors calculus course at harvard in the 1960's, and stewart was written for a non honors calculus course at the university of georgia in the 1990's.

i.e. spivak is on another plane disjoint from (and higher than) stewart.