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Analysis 2 Textbook?

  1. Oct 1, 2009 #1
    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?
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  3. Oct 1, 2009 #2
    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.
    Last edited by a moderator: Apr 24, 2017
  4. Oct 2, 2009 #3


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    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.)
  5. Oct 2, 2009 #4
    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?
  6. Oct 2, 2009 #5
    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.
  7. Oct 11, 2009 #6
    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?
  8. Oct 12, 2009 #7


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    If you have read and understood Courant volume 1, you should be in very good shape to tackle Rudin.
  9. Oct 12, 2009 #8


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    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.
  10. Oct 12, 2009 #9
    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.
  11. Oct 12, 2009 #10
    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".
  12. Oct 12, 2009 #11
    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?
  13. Oct 12, 2009 #12
    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."
  14. Oct 12, 2009 #13


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    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.
  15. Oct 12, 2009 #14
    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?
  16. Oct 12, 2009 #15


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    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.
  17. Oct 12, 2009 #16
    Hmmm...If I go ahead with Rudin, do you think I will be prepared for the course next fall?
  18. Oct 12, 2009 #17


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    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.
    Last edited: Oct 12, 2009
  19. Oct 12, 2009 #18
    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!
  20. Oct 12, 2009 #19


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    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.]
  21. Oct 12, 2009 #20
    Would you say there is more overlap between Munkres and Pugh than Munkres and Rudin?
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