A bit of history about Calculus

Discussion in 'Math' started by amilton542, Dec 10, 2014.

  1. amilton542

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    Does anyone know what the story is with Bernhard Riemann and Calculus? He seems to have achieved a lot of merit for the fundamentals of the discipline; in particular the fundamental theorem of calculus. I've always believed it was Newton and Liebnitz who made these important discoveries?
     
  2. studiot

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    That is why we call it the Riemann integral.
     
  3. Papabravo

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    And it is only one of about a dozen different types including my personal favorite: the Ito integral.
     
  4. amilton542

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    Yes, there are lots of different names for a variety of integrals but you still didn't answer my question. I'd imagine Newton's and Liebnitz contribution was only to do with tangent problems, correct?
     
  5. studiot

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    Newton and Leibnitz did not work with what we now formally call functions. They worked with what we now call plots and with tables of values, ie points.
    They introduced differentiation and integration as a limiting process.
    Their integral, in particular we now call the indefinite integral.

    Riemann, coming later worked with the idea that an entity can have a separate existence of its own, that is a table of x values, and correspondence to a table of y values.

    That is the whole curve of (x,y) is a set of points we call the function f(x).

    If we collect together a set of all functions with a common property we can imagine a mapping or function that connects each of these functions to another set of functions in some way, but uniquely.

    Differentiation is such a connection, but indefinite integration is not unique.

    Rieman introduced the idea of a unique connection between two sets of functions


    I\left\{ {f(t)} \right\} = \int\limits_0^x {f(t)dt}

    Which provides an output function from the second set, on input from the first set.

    Differentiation can be handled in the same way, by using the D operator

    So the derived function (or derivative) of a function f(x) = f'(x) is another function in its own right


    D\left\{ {f(t)} \right\} = f'(t)

    This then allows the introduction of the fundamental theorem of calculus that basically joins all these sets of functions into one set that now constains the function, its derivative and allows us to select the derivative and the integral as inverse functions for any given function f(x) in the set.

    This then formed the basis for transform theory

    Sorry if this dashed off piece is a bit rambling ask for clarification if needed.
     
  6. Papabravo

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    It is also important to remember that all of this innovation happened over a period of approximately three centuries. It did not spring into being overnight. An amazing result of the process of stepwise refinement.
     
  7. amilton542

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    No not at all. I like listening to someone rambling on, intellectually that is. Probably why I'm still single. No issues there, that was great. Thanks.
     
  8. studiot

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    It is important to realise that there are a lot of ifs and buts to my very broad brush generalisations.

    These were meant for understanding only. You need to study the ifs and buts carefully to use these ideas in detail.
     
  9. amilton542

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    The only thing I didn't get was how it formed the basis for transform theory?
     
  10. studiot

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    I can't say it better than Mr Churchill. My previous post was really by way of commentry on his excellent introduction.

    TF1.jpg
     
  11. amilton542

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    What book is that? I'm a bit of a fussy bugger when it comes to books but I always like what you recommend. Tell me :)
     
  12. studiot

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    Ruel V Churchill, Late professor of Mathematics at the University of Michigan has written three 'companion' books covering much engineering maths

    Operational Mathematics (originally Modern Operational Mathematics for Engineering) which is largely about the Laplace transform

    Complex Variables and Applications

    Fourier Series and Boundary Value Problems

    The excerpt came from the first.

    Our own Professor Ian Sneddon, late of Glasow University has writeen three similar texts

    Fourier Transforms
    A massive tome 90% applications 'taken from real recent papers' rather than over simplifed examples

    Fourier Series

    Elements of Partial Differential Equations
     
    Last edited: Dec 11, 2014
  13. Papabravo

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  14. amilton542

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    I don't think I'll be buying any latest hardcover edition for a ridiculous amount of money. I've learned to tame my temptation in that department. There's a few used former editions on Amazon going for about £4 or so, they'll suit me fine for now.
     
  15. studiot

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    You'd be better off buying shares in __zon at those prices.
     
  16. amilton542

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    Shares in Amazon? They should me owe free Christmas gift vouchers for all the hardcovers I've paid for.
     
  17. bogosort

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    Hi, both Newton and Leibniz understood and used the definite integral; in fact, the definite integral was in regular use before their time. The reason Newton and Leibniz are given credit for being the fathers of calculus is because they were the first to fully work out the significance of the inverse relationship between differentiation and integration. They did not, however, understand the concept of a limit and so couldn't provide a mathematically rigorous foundation for their work. (That would start in earnest a century later with Cauchy.)

    Riemann's primary contribution to calculus was the work he did in formalizing what exactly an integral is, and what kinds of functions can be integrated (a surprisingly small class of functions, which was later remedied by Lebesgue). I don't think professional mathematicians consider Riemann's work on integrals particularly important, though his "geometrically-friendly" approach is certainly helpful to first-year calc students.
     
  18. studiot

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    Welcome bogosort and thank you for your input, hopefully the first of many useful ones.

    I did say my treatment was elementary broad brush and you might be interested in more detail

    http://www.sjsu.edu/faculty/watkins/infincalc.htm

    In fact both Leibnitz and Newton knew the key point underlying Limits, and one convergence test even bears Leibnitz' name, although their approach was different from either of the modern ones.

    The key point is, of course, convergence and both used this in their approaches.

    Newton's approach was via what we now call finite differences, Leibnitz via infinite series.

    Both were working with individual geometric points, today we work with functions.
     
  19. bogosort

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    Hi, thanks for the warm welcome!

    I'm a math history geek but I'm certainly not a scholar, so I could well be wrong. But while I wholeheartedly agree that both Newton and Leibniz were masters of infinite series and had an intuitive handle on convergence issues, I cannot agree that they understood the "key point underlying limits". When challenged to explain their mystical vanishing quantities or infinitesimals-- and they were often challenged, most famously by their contemporary Bishop Berkeley, who took Newton to task for his nonsensical "ghosts of departed quantities" -- neither provided any clear indication of a mathematical limiting process. Newton actually uses the word limit when describing his "ultimate ratios" of "evanescent quantities", but he does so in physical terms, meant to evoke an intuitive understanding from the contemporary reader. For Newton, the limit is simply the end result, not a mathematical tool. His vanishing quantities were thus left unexplained. Likewise for Leibniz, infinitesimals were a thing onto themselves, mathematical objects in their own right and not needing of a limiting process.

    D'Alembert (teacher of Laplace) rebuked Newton and Leibniz for their metaphysical mumbo-jumbo: "A quantity is something or nothing; if it is something, it has not yet vanished; if it is nothing, it has literally vanished. The supposition that there is an intermediate state between these two is a chimera.” In 1754, in an article for the Encyclopedie (of which he and Diderot were editors), d'Alembert published the first reasonable definition of limit: "One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude, however small, though the second may never exceed the magnitude it approaches." Hardly epsilon-delta, certainly, but finally on the right track. Using this he defined the derivative in terms of the limit of a difference quotient, so familiar to us now. It would take another century before Cauchy significantly improved this definition (the great Euler simply waved his hand and let dx = 0 when it was convenient!) , and half a century after that for Weierstrass to provide us with the rigorous definition we go by today.

    In any case, Newton's mathematical genius is unquestioned and perhaps unrivaled. It matters not whether he understood the limit process on an intuitive level that he could not verbally describe, or if he simply had no need for limits (see non-standard analysis for a modern take on calculus sans limits). Ultimately, history makes clear that calculus was invented by humanity; it is far too deep for any one person, even a giant such as Newton.

    Of course this is history, so there is no Truth with a capital T; alternative interpretations are welcome!
     
  20. studiot

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    Before you pass judgement on Newton, you should perhaps read this

    http://www.jstor.org/discover/10.2307/229868?sid=21104953194321&uid=4&uid=2&uid=3738032

    I suggest you are selling both Newton and Leibnitz short in their grasp of limits.

    The interesting thing is that we are moving steadily away from Vector & Tensor Calculus towards the use of Differential Forms, which are a 'Calculus' that harps back to Leibnitz Theorem that the cross products (second order and greater) of small quantities vanish. Basic details of this were introduced in the article linked to in my previous post.
    Remember also that Newton was primarily a Physicist who wanted to calculate results (hence the term calculus) and, as I said earlier, had already developed the calculus of finite differences, and worked with tables of quantities.
    I do not know if you are old enough to remember using these?
    When I started if I wanted accurate, I used Chambers 7 figure tables and fintite difference interpolation techniques (mostly due to Newton).

    see also

    http://www.jstor.org/discover/10.2307/229868?sid=21104953194321&uid=4&uid=2&uid=3738032

    He did indeed understand the distinction between the very small and the what we now call the derived function in his calculus of fluxions and fluents, however this produces an indefinite integral, as I previously noted.
     
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