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Calculus or Analysis?

  1. Mar 31, 2012 #1
    I came across a copy of the "Advanced Calculus Problem Solver" for super cheap in a thrift shop. (Along with a couple other gems, like one on Fourier analysis!) It's a series I like to use, and I had the regular "calculus" one (well, I have the libraries copy).

    It's from 1981. I thought it'd be useful for Calc III, bu the first chapter is "point set theory," and the first problem is "Show that the set q of rational nmbers x such that 0 < x < 1 is countably infinite." The rest of the chapters include Vector spaces (2) Continuity (3) Elements of partial differentiation (4).

    Some of which I'm sort of familiar with - but it's written in a very proofy-symbolic-general kind of way and not what I'm seeing in my calculus book. Is this Analysis? I'm afraid I'm not far along enough to quite understand the difference yet.

    -Dave K
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  3. Mar 31, 2012 #2


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    It sounds like what you have is an analysis book. Analysis at that level is essentially proving all the results that you use in calculus class (fundamental theorem of calculus, intermediate value theorem, etc)
  4. Mar 31, 2012 #3
    Thanks. I think I'm getting the picture now. Good find for $1.00.
  5. Mar 31, 2012 #4


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    Sounds like the type of stuff that would normally be covered in a course with "analysis" in the name. But calculus is really just "analysis dumbed down". (Micromass said something like that to me once, and I think I agree). I guess another way of saying that is that analysis is calculus done right. So the book should definitely be useful, but it may prove a few results that won't be covered by your "Calc III".

    (I'm not familiar with the standard courses in the USA, so I don't know what Calc III contains).
  6. Mar 31, 2012 #5
    Calc III is when we start getting into multiple variables and now, towards the end, we are starting to cover double integrals. Syllabus here: http://math.usf.edu/ug/syllabi/mac2313/

    I like that you guys are clarifying Calc vs. analysis for me. Thanks. As much as I love my university, it is rather large so it's hard to find guidance on this stuff. My advisor was not even a math major!
  7. Mar 31, 2012 #6
    There is a bit of a historical mismatch in course names. Many "Advanced Calculus" courses of past years would now be called Analysis.

    Take a look at Loomis & Sternberg here:

    The way I look at it, the Advanced Calculus course I did (and the books I have seen) are just the merging point between upper-level vector calculus and rigorous analysis. You might also call it "Applied Analysis", but there are other things that could take that name, as well.
  8. Mar 31, 2012 #7
    That reminds me of another thing I was wondering about. Here's the sequence at my school: http://i40.tinypic.com/1zwge54.jpg (I've spent hours staring at this thing mapping my future).

    You'll see that under analysis you can take either Intermediate Analysis or Vector Calculus. Back when I thought I was going in a more physics-y dimension I was pretty sure I was going to take Vector Calc, but now that I'm in a more math direction, and after being advised "you haven't really done a math degree if you don't take analysis" I'm taking that.

    What are the overlaps and potential gaps? (Of course I can still take both).

    The old flow chart had the "Intro to real analysis" following *either* course, but this latest one has only Intermediate Analysis allowing one to take Real analysis.

    Shortened version of the question if this is to blabby: What is the difference between vector calculus and Real Analysis?

    -Dave K
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