Next Book for Analysis: Courant's Volume 2 or Munkres' Analysis on Manifolds?

[JG89]

I've finished Richard Courant's "Introduction to Calculus and Analysis" Volume 1 (single variable calculus), and I was wondering, what is the next book that I should read? I was thinking of his volume 2 which deals with multi-variable calculus, but there is a math course I will have to take next year that uses Munkres' "Analysis on Manifolds", so I was thinking I should use that book.

How do you guys rate Analysis on Manifolds?

 

[n!kofeyn]

I've only browsed through the book when I was taking a more abstract calculus on manifolds course (we used Introduction to Smooth Manifolds by John Lee). I would say that the notation and approach was different enough that it wasn't helpful, but the differences in the type of course each book was written for needs to be taken into account.

A book that I would recommend going through is Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by John Hubbard. I've only read the excerpts (http://matrixeditions.com/UnifiedApproach3rd.html"), but it looks to be a fantastic book. In particular, it relates the standard vector calculus to the differential form approach, which to my knowledge is not done in detail in very many books. The book appears very comprehensive and lucid, so I think it would probably make the Munkres seem much easier if you went through it first without the pressure of a class. Although, Munkres does usually write at very readable level.

 

[Landau]

I think Analysis on Manifolds (the verbose version of Spivak's Calculus on Manifolds) would be an excellent follow up, certainly since you are taking a course based on the book next year. But you will need some linear algebra for that book, so if you haven't done any linear algebra yet you should fix that first. (Axler's Linear Algebra Done Right is an option.)

 

[JG89]

For LA I've done the basics: vector spaces, linear transformations, systems of equations, matrix algebra, inner products.

Is this a sufficient background in LA?

 

[n!kofeyn]

You mainly need to be comfortable with linear transformations, the determinant and its properties, column/row rank of a matrix, being able to determine whether a matrix is injective/surjective, basis of a vector space, linear independence, and proving that a linear transformation is injective/surjective. Of course, some of this may be reviewed in the class, but it may be assumed that you know most of it.

 

[JG89]

I was thinking about Rudin`s Principles Of Mathematical Analysis, but I am not sure if I have all the prior knowledge required to study from that book.

What do you guys think? What prerequisites should I satisfy before I study from his book?

 

[jbunniii]

I was thinking about Rudin`s Principles Of Mathematical Analysis, but I am not sure if I have all the prior knowledge required to study from that book.

What do you guys think? What prerequisites should I satisfy before I study from his book?

If you have read and understood Courant volume 1, you should be in very good shape to tackle Rudin.

 

[MathematicalPhysicist]

It really depends, the textbook by Munkres is a modern textbook on calculus and it covers other material than Courant's second volume.
From my memory, Courant covers in the second volume Lagrange multipliers (I can't recall Munkres covering this topic) also topics in complex analysis, but mainly this textbook is about classical multivariable calculus, and he doesn't use the modern manifolds theory, wheras Munkres' textbook is mainly using the modern ideas of manifolds, Stokes' general theorem is covered in Munkres but not in Courant, and there's a brisk overview of DeRham Cohomology wheras Courant doesn't.

If you just covered Courant first volume then I think the next textbook should be his second volume and you can supplement it with Munkres for the more modern account.

 

[n!kofeyn]

I am confused JG89. You first asked about where to move onto next because you are taking a course that uses Analysis on Manifolds by Munkres next year (next semester or next fall?), and you've finished Courant's first volume. Now you're asking about Rudin's analysis book. I assume you're probably a 3rd year student, so aren't you probably taking an analysis course right now? Rudin's book is by far not the best option, especially pedagogically. Mathematical Analysis by Apostol is better, though pretty dry. There are others.

But what is it that you're wanting advice on exactly? You don't necessarily need to know a lot of analysis for the Munkres book, as I think a better name for the book is calculus on manifolds. The Munkres book is basically teaching an abstract and generalized calculus, and Courant's 2nd volume teaches the standard multivariable calculus.

The reason I mentioned the book by Hubbard is that it will cover both the material that Courant covers AND some of the material that Munkres covers. It's sort of a multivariable/vector calculus, analysis, linear algebra, and calculus on manifolds book all in one.

 

[qspeechc]

Just to throw in another suggestion: "Advanced Calculus: A Differential Forms Approach" by H. Edwards, something like the Hubbard book, but a bit more advanced.
Or if you're looking for an alternative to baby Rudin I can suggest Pugh's "Real Mathematical Analysis".

 

[JG89]

Thanks for the suggestions guys. I've heard good things about baby Rudin and Pugh's analysis book. I'm surprised to know that Courant Volume 1 is sufficient to study from baby Rudin. I've always thought that was quite an advanced book.

I'm really torn between picking Munkres' Analysis on Manifolds and baby Rudin. I've looked at the table of contents for Pugh's book, and it seems like baby Rudin is more similar to Munkres' book than Pugh's book is. Also, I'm going to have to use Analysis on Manifolds for a course next year, so I probably won't be using Pugh.

Do you think the fact that the course I'm taking next year teaches out of Munkres should hold enough weight for me to use Munkres' book instead of Rudin's?

 

[JG89]

Now that I read more reviews and compare the table of contents more closely, it doesn't seem like I'm giving Pugh's book enough of a chance. Still, I'm having a hard time deciding which textbook to get...

For what it's worth, here's the description for the course that I will be taking: "Topology of Rn; compactness, functions and continuity, extreme value theorem. Derivatives; inverse and implicit function theorems, maxima and minima, Lagrange multipliers. Integrals; Fubini’s theorem, partitions of unity, change of variables. Differential forms. Manifolds in Rn; integration on manifolds; Stokes’ theorem for differential forms and classical versions."

 

[Landau]

Rudin and Pugh are different books, I don't think they are appropriate for the course, they focus on different topics. You'll really want to have a book like Munkres, which perfectly fits the course description. Alternatives are Spivak's Calculus on Manifolds, and Loomis and Sternberg, the latter is for free available here.

 

[JG89]

Sorry for the confusion. I'm a first year student, just started this fall, but I covered Courant Volume 1 last year. So everything I'm doing in my calculus course right now (teaches out of Spivak) is more or less review, which is why I wanted to read ahead with another book. The course which uses Munkres is next fall. I just asked about baby Rudin because I wasn't sure if I was ready to read it or not yet. I guess I will stick with Munkres, and then after that I will be ready for Rudin?

 

[Landau]

It wasn't clear to me what your goal is. If you want to learn the material your course covers, then get Munkres (or the ones named above, Spivak, L&S). If you just want to learn 'more analysis', then Rudin, Pugh, Apostol are also all good. I'm pretty sure you're already ready for all of them, since Courant's book is a great preparation.

In short: after a decent first introduction to analysis (Courant, Spivak's or Apostol's Calculus,...) both Rudin and Munkres are ok as the next step. They just cover different topics / have a different focus.

 

[JG89]

Hmmm...If I go ahead with Rudin, do you think I will be prepared for the course next fall?

 

[Landau]

Rudin's Chapter 9 (Functions of several variables) and 10 (Integration of Differential Forms) overlap with your course. So doing all of Rudin will certainly prepare you for the course. Obviously, 'being prepared' is not the same as 'knowing all stuff already beforehand'.
Still, you need to know linear algebra, especially the topics named in message #5.

 

[JG89]

Ahh I'm so torn between picking Munkres or Rudin. Munkres is more "necessary" but I think I will have more fun with Rudin.

Thanks for your help Landau!

 

[Landau]

You'll need Munkres anyway next year, so take Rudin now ;)

[Beware that Rudin is dry, and overpriced. Personally I like Apostol and Pugh a lot. There's just too much choice between good books I guess. Going to your library might help to decide.]

 

[JG89]

Would you say there is more overlap between Munkres and Pugh than Munkres and Rudin?

 

[jbunniii]

Would you say there is more overlap between Munkres and Pugh than Munkres and Rudin?

Not appreciably more, no. Rudin and Pugh cover quite similar material.

I own all three books, and I would characterize Munkres as a much-expanded treatment of the material covered in either Pugh chapter 5 ("Multivariable Calculus") or Rudin chapters 9-10 ("Functions of Several Variables" and "Integration of Differential Forms").

As an aside, I recommend finding a used hardcover copy of Munkres. The current in-print edition is a crappily reproduced paperback, which is outrageous given the $76 list price.

 

[n!kofeyn]

I don't understand why you're insisting Rudin JG89. It is by no means the best book on analysis. Since you are taking a course using Munkres in the future, then I would recommend using a different book before that course. The reason is because it is always nice to see a subject from different perspectives and viewpoints. For this I would still recommend the book by Hubbard, as it contains all the relevant prerequisites and has around a 100-page index of analysis proofs. It isn't a basic book by any means, and although I haven't read it (I've only browsed through it reading bits and pieces), it may even be more comprehensive than Munkres' book. I also recommend the Advanced Calculus book by Edwards that was mentioned above. It is more nonstandard, but is very readable. I wish I had read it before I took my course in calculus on manifolds.

A book like Rudin will, in my opinion, not prepare you for the Munkres course as well as the two books above. They are very intuitive books, but advanced, and they will be more suited to the material covered in Munkres' book (which isn't the best book for that material in my opinion). I find Rudin to be a terrible reference book and slyly written. If you want learn analysis, then I recommend using something other than Rudin. Do you know how to do proofs? Courant's book is really just a calculus book, and not really an abstract mathematics book, so you need to be able to do proofs before you attempt a course in calculus on manifolds. I might recommend Analysis by Steven Lay because it has a good introduction to doing proofs and introductory analysis (it is an easy book). Then you could move on to the advanced calculus books mentioned.

 

[jbunniii]

Thanks for the suggestions guys. I've heard good things about baby Rudin and Pugh's analysis book. I'm surprised to know that Courant Volume 1 is sufficient to study from baby Rudin. I've always thought that was quite an advanced book.

No, Rudin is not as advanced as you think. There is actually considerable overlap between Rudin and either Courant or Spivak, both of which are quite good preparation for Rudin.

Rudin implicitly assumes that you have achieved some mathematical maturity with rigorous calculus, including epsilon/delta arguments: essentially the level of maturity that you get from working through Courant or Spivak.

Rudin treats the same calculus material a bit more abstractly, which allows him to prove familiar results in greater generality than in Courant or Spivak.

For example, Rudin increases the level of abstraction slightly by introducing metric space topology. As you will see, this actually cleans up and generalizes some of the messier epsilon-pushing proofs from Spivak or Courant.

For example, Spivak has a chapter called "Three Hard Theorems" having to do with what you can say about continuous functions on a closed interval. In Rudin, these are proved for continuous functions on compact, connected sets in a metric space. This has the advantage of making the proofs considerably easier/cleaner while also extending the applicability of these theorems to more general settings, e.g., to R^n.

Another difference is that Rudin works with the Riemann-Stieltjes integral (of which the Riemann integral is a special case), which is neat but not really what I would call essential for a first course in analysis. It's not much harder than standard Riemann integration, though, and it's useful grounding for later on when you study Lebesgue integrals.

Finally, Rudin has two dense chapters on multivariable calculus and one even denser chapter on Lebesgue integration. You would be better off reading other books for this material: Munkres' "Analysis on Manifolds" for the multivariable stuff and Bartle's "The Elements of Integration and Lebesgue Measure" are good choices.