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What is the last level of calculus?

  1. Oct 24, 2004 #1
    What is the "last" level of calculus?

    Well, call me a geek, nerd, freak, whatever, but I absolutely love math. Especially calculus.

    I am 16 and am so far in training to become a professor in physics and/or quantum theory at a major university. I may also teach advanced calculus.

    My question is..what is the boundry at where calculus "ends"? Of course, it never "ends", as you can always learn more. But..what is the most you'll need to know to pretty much learn every field in science?

    So far, I'm up to vector calculus. I just heard of Fourier Series..it's too advanced for me so far. Is there anything past this? Heh, fourier series seems complicated enough..

  2. jcsd
  3. Oct 24, 2004 #2


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    Personally you might know certain areas, but at the same time you really don't know those areas.

    If I were you, I would focus on limits and truly understanding where calculus comes from. After that, you can move on.

    Calculus seems to never end, and that's good because we all love it.

    Note: I just want to be a professor at any canadian university. I just want to do what I love, regardless of pay or fame. :)
  4. Oct 24, 2004 #3
    Luckily it never ends, just change name and form, it expands. Calculus, Analysis, Complex Variable, ODE's, PDE's, Calculus of Variations, Differential Forms.... I could just go on and on and on :biggrin:
  5. Oct 24, 2004 #4
    Fourier is not so hard in calculus. I think the complex variable theory is harder. But also is not at the top. We can not set a point that we can say: there's the end of the way of our knowledge, because that would be false.
  6. Oct 24, 2004 #5


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    The end hasn't been found yet, because diff eqs are still a part of calculus (what else?) and research on them is still going on with new papers posted on the arxiv every day.
  7. Oct 24, 2004 #6


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    Actually computing the terms of a Fourier series isn't bad. You just have to be up on your trigonometric integration. Once, you're familiar with what, it's just doing integration and adding the terms up, and simplifying. What's difficult about the Fourier series is understanding it's properties, like why you can add all the "infinite terms" - it'll "become" that function that you're doing the series on. This is mainly because of orthogonal nature of the Fourier series.

    As someone said earlier, you need to be very confident with limits. I or anyone else, cannot stress how important limits are to the calculus. The conceptual jump from elementary algebra to elementary calculus is the idea of the limit. Many things in calculus cannot be without that of the limit. I just used an example of a limit above with the Fourier series.

    From what I've gathered during my education, most people seem to think that Calculus ends after an elementary Calculus sequence (including DE). Many courses afterwards involving the Calculus, focus on proving and figuring out why things work that you or I took for granted during our sequence of elementary Calculus courses. There's all sorts of combinations and additions people make to Calculus - Complex Variables, basically studies the calculus on complex-valued functions.
    Last edited: Oct 24, 2004
  8. Oct 24, 2004 #7
    Heh, the weird integral sign with a circle though it gets me..

  9. Oct 24, 2004 #8


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    The "weird integral sign with a circle" is something called a "closed integral." Closed integrals denote integrating over a closed surface, like a circle.

    Like I said, I strongly suggest you become very familiar with limits. Integrals and derivatives are only conceptually understood through their limit definitions. From there, you'll be able to become familiar with the techniques of integration, and then you'll be absolutely ready for vector and multivariable calculus.

    I was in the same boat as you once, I had it embedded in my mind that I'd be alright without a firm understanding of limits - boy, was I wrong.
  10. Oct 25, 2004 #9
    Are you studying on your own? If so, what materials are you using? Are you certain that you have a very solid foundation on the basics?

    I studied calc on my own, and while helpful, I later found that my zeal for wanting to advance and my neglect of rigorous study of the fundamentals later came back to bite me.

    I mean, sure, you can read through a calc text book in a couple of days, and maybe even work the example problems and get the same answer, but if you can't explain why the answer is what it is, and explain what the answer means, it's really just a hollow, superficial knowledge of the subject.
  11. Oct 25, 2004 #10

    Exactly, and this is why I make sure I UNDERSTAND the CONCEPT of each problem. If I don't, I go back to limits and review or keep reading the problem.

    I understand integration alot more than I do limits. Intregration makes a TON of sense to me. (That is, the basics..)

    if I could figure out how to write the integral sign from a to b f(x) dx = some derivative of L.
    (dont understand the code that well yet)

  12. Oct 25, 2004 #11
    I'm learning calc on my own. And let me tell you; it's very tedious. I have trouble with comphresion to boot, I'm visual.

    Recently I moved to a new school, called Desert Technology. EVERYTHING is on computers, the school is considered a new school of its kind. We use top of the line computers, and every subject is on the computers. Even math.

    However, its a brand new school and very small at 300 students, the highest math they teach so far is plain old geometry.

    Like I said, I want to be a professor. I want to for a good reason; I'm great at teaching.

    Tommarrow (monday) I will be spending an extra 4 hours of my time after school (I won't get paid; it's for credits) helping sophmores and freshmen with math as a tutor.

    I love to teach :)
  13. Oct 25, 2004 #12
    You may enjoy a book that I have called "How to Ace Calculus: The Streetwise Guide". It's a good companion book and it's only like $15. I think that it does a good job of explaining limits.

    One other thing. If the school that you will attend is anything like mine, your calculus class will be slow. Very slow. We started on August 20th and we're like halfway done with chapter two.
  14. Oct 25, 2004 #13


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    But, see that's the thing. The integral can only be understood conceptually from the limit of something called a Riemann sum. Applying the power rule or doing some simple subtitution is integrating, yes, but it's just simple algebra. Calculus *is* the limit.
  15. Oct 25, 2004 #14


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    I can't imagine writing a proof in Calculus without the knowledge of limits.

    Also, limits are not as easy as everyone thinks. Sure after awhile it gets easy, but not at the beginning because there is so much you don't know.
  16. Oct 25, 2004 #15
    Integration has a lot of sense if you talk about real domain. When you get in complex variable integration... the thing changes a lot
  17. Oct 25, 2004 #16


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    The concept of the limit isn't so bad, it's being able to evaluate the limits. Which that is part, intuition, and part, algebra - L'Hopital's rule aside.
    Last edited: Oct 25, 2004
  18. Oct 26, 2004 #17
    Complex variable theory is, in my opinion one of the most elegant branches of mathematics. I took two courses in complex variables in college, and loved them both!
  19. Oct 26, 2004 #18


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    You're getting me excited! :blushing: :rofl:

    I'm reading a book on complex numbers, and I should be hitting an introduction of Complex Variables on the last chapter. After that, I'll go on from there and hope for the best.

    The most amazing thing is...

    de Moivre's Theorem (or Euler's Formula)

    ...that I have seen so far. I hope there is much more to go.

    Before e used to be a lame vowel, but after calculus and complex numbers I'm just like :surprised whenever it comes up.
  20. Oct 26, 2004 #19
    I second that. Complex analysis is so strange it's beautiful.
  21. Oct 26, 2004 #20
    We still need to solve those differential equations, too.
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