Results 1 to 14 of 14

Math Help - What is analysis?

  1. #1
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,823

    What is analysis?

    If someone had never taken Analysis before, but asked you what it was, what would you say?

    A friend of mine said that you take everything you ever learned about math, and then relearn it, but this time, leaving NO stone left unturned. Is this a good way to look at it?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Jul 2009
    Posts
    152
    When I was coming up, "analysis" just meant Calculus. Thus "complex analysis" is the Calculus of complex-valued functions, etc. "Real analysis" deals in an involved way with the topology of the real numbers in order to rigorously construct the apparatus of Calculus, etc... If one says that Liouville used "analytical methods" to prove the existence of transcendental numbers whereas Cantor used set theoretic arguments, one is referring to the fact that Liouville's proof uses the Mean Value Theorem for Derivitives from Calculus, etc.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Nov 2006
    From
    Florida
    Posts
    228
    Quote Originally Posted by AlephZero View Post
    When I was coming up, "analysis" just meant Calculus. Thus "complex analysis" is the Calculus of complex-valued functions, etc. "Real analysis" deals in an involved way with the topology of the real numbers in order to rigorously construct the apparatus of Calculus, etc... If one says that Liouville used "analytical methods" to prove the existence of transcendental numbers whereas Cantor used set theoretic arguments, one is referring to the fact that Liouville's proof uses the Mean Value Theorem for Derivitives from Calculus, etc.
    This pretty much the explanation I give as well. "An advanced, proof based treatment of all you learned in calculus and then some." Though this changes depending on who I am talking to of course.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,823
    Quote Originally Posted by putnam120 View Post
    ...Though this changes depending on who I am talking to of course.
    You're talking to a guy that has a good hanlde on Calc that's trying to understand what to expect for Analysis.

    Two questions:

    1. Would it be better to take DEs or Linear algebra before analysis? Or the other way around?

    2. What is the absolute best book for beggining analysis (real)?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Jul 2009
    Posts
    152
    Quote Originally Posted by VonNemo19 View Post
    You're talking to a guy that has a good hanlde on Calc that's trying to understand what to expect for Analysis.

    Two questions:

    1. Would it be better to take DEs or Linear algebra before analysis? Or the other way around?

    2. What is the absolute best book for beggining analysis (real)?
    1. I can't think offhand of a reason why you'd need either ODE or linear algebra for real analysis, although you might want to take linear algebra first because it's usually an easier course. ODE requires some linear algebra (Wronskians, etc.), so take linear before that.

    2. There's probably never going to be a concensus on what's the "best" book on real analysis. I'm partial to Richard R. Goldberg's Methods of Real Analysis (2nd ed., John Wiley & Sons, 1976), because that's the one I learned it from, though I'm a broken down old dude.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member McScruffy's Avatar
    Joined
    Jul 2009
    Posts
    87
    Awards
    1
    Quote Originally Posted by VonNemo19 View Post
    ...Would it be better to take DEs or Linear algebra before analysis? Or the other way around?...
    What about complex vs. real analysis in terms of which should be taken first?
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Jul 2009
    Posts
    152
    Quote Originally Posted by McScruffy View Post
    What about complex vs. real analysis in terms of which should be taken first?
    I don't think it makes an awful lot of difference. I guess I would say take real analysis first, because it deals intensively with strict definitions of continuity, sequences, and series. That way, when issues of continuity and such come up in complex analysis, it will be really clear to you what's going on. In my experience, the really hard stuff in complex analysis is in two areas: contour integration and residue theory. I suppose you'd be somewhat better prepared for each after having the kind of ultra-rigorous integration and infinite series work one usually gets out of a real analysis course.
    Last edited by AlephZero; July 25th 2009 at 08:54 PM. Reason: typo
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by VonNemo19 View Post
    If someone had never taken Analysis before, but asked you what it was, what would you say?

    A friend of mine said that you take everything you ever learned about math, and then relearn it, but this time, leaving NO stone left unturned. Is this a good way to look at it?
    Analysis is the rigorous study limiting processes over various topological spaces, for real analysis these spaces will be the reals and assorted function spaces over the reals, for complex analysis ...

    All you learned in calculus will be re-done in real analysis but without missing out the basic assumptions that you probably never noticed that you had to make.

    The main reason that analysis emerged in its present form was to make sense of Fourier series (and to cure the problems that (George) Bishop Berkeley identified in "The Analyst"). You might be interested in "A Radical Approach to Real Analysis" by David Bressoud which discusses the emergence of analysis.

    Note before the emergence of modern analysis in the early 19th century the term analysis referred to calculus.

    CB
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Member
    Joined
    Nov 2006
    From
    Florida
    Posts
    228
    Quote Originally Posted by VonNemo19 View Post
    You're talking to a guy that has a good hanlde on Calc that's trying to understand what to expect for Analysis.

    Two questions:

    1. Would it be better to take DEs or Linear algebra before analysis? Or the other way around?

    2. What is the absolute best book for beggining analysis (real)?
    1. I personally took Linear algebra before analysis but that was mainly because it was a prereq. for the class. Really I found it unnecessary. However, it would have been nice to know Linear algebra if we had studied functions f:\mathbb{R}^n\to\mathbb{R}^m. As for DEs I found that some results in my DE classes made more sense after taking analysis, but that's just me. Though I would suggest taking an "introductory" DE class after your initial calculus sequence.

    2. This really depends on the person. I personally liked Rudin (my first analysis book) once I got past the topology chapter. However, I know people who hate it.
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor
    Jester's Avatar
    Joined
    Dec 2008
    From
    Conway AR
    Posts
    2,338
    Thanks
    19
    Quote Originally Posted by VonNemo19 View Post
    You're talking to a guy that has a good hanlde on Calc that's trying to understand what to expect for Analysis.

    Two questions:

    1. Would it be better to take DEs or Linear algebra before analysis? Or the other way around?

    2. What is the absolute best book for beggining analysis (real)?
    On your first question, I would recommend taking both before real analysis. Real analysis is a sophisticated subject and the more mathematically mature you are the better you will be prepared. Just my two cents.
    Follow Math Help Forum on Facebook and Google+

  11. #11
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,605
    Thanks
    1573
    Awards
    1
    Quote Originally Posted by Danny View Post
    On your first question, I would recommend taking both before real analysis. Real analysis is a sophisticated subject and the more mathematically mature you are the better you will be prepared. Just my two cents.
    I generally agree with that statement.
    In my experience, the main value in undergraduate ‘Complex Variables’ lies in its teaching the nature of a complex numbers. (That is not done completely in any other undergraduate course.)
    Now turn to Linear Algebra. In Linear Algebra one learns about basis. One learns to use matrices to for a many different purposes.

    Both of those out comes will enhance your study of real analysis.
    Follow Math Help Forum on Facebook and Google+

  12. #12
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,823
    Thanks guys. (and girls if there are any)
    Follow Math Help Forum on Facebook and Google+

  13. #13
    Senior Member Sampras's Avatar
    Joined
    May 2009
    Posts
    301
    Here is a good description of the "point" of doing real analysis: Here.
    Follow Math Help Forum on Facebook and Google+

  14. #14
    No one in Particular VonNemo19's Avatar
    Joined
    Apr 2009
    From
    Detroit, MI
    Posts
    1,823
    Quote Originally Posted by Sampras View Post
    Here is a good description of the "point" of doing real analysis: Here.

    Very good!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help with Analysis please :)
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: October 16th 2009, 04:38 AM
  2. analysis help
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: October 8th 2009, 03:11 PM
  3. analysis help
    Posted in the Calculus Forum
    Replies: 0
    Last Post: January 19th 2009, 02:38 PM
  4. analysis
    Posted in the Calculus Forum
    Replies: 2
    Last Post: December 15th 2008, 03:07 PM
  5. Analysis Help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: September 8th 2008, 05:59 PM

Search Tags


/mathhelpforum @mathhelpforum