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Touring the Calculus Gallery: The Cauchy and Modern Wings

  • The Corner Bakery Cafe

    17th and JFK Blvd, Philadelphia, PA (map)

    39.954006 -75.168213

  • The entrance is at the corner of 17th St. and JFK Blvd., one block West and one block South of Capriccio's. We will be in the large area in the back.
  • Price: $1.00 /per person

    Refund policy

  • Let us stroll through the history of calculus, review some of its modern techniques starting with the rigors of the limit, and roll up our sleeves to do some exercises. We will briefly explore the work of Augustin-Louis Cauchy (1789­-1857), Karl Weierstrass (1815-­1897), Georg Friedrich Bernhard Riemann (1826­-1866), Vito Volterra (1860­-1940), Georg Cantor (1845­-1918), René Baire (1874-­1932), and Henri Lebesgue (1875­-1941). We will discover some of the ideas that helped make modern analysis an icon of sophisticated mathematically exact thinking.

    "I think it [the calculus] defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking." — John von Neumann

    Our exploration will discuss (with a focus on the second half: pages 10-19, sections 7-10, but we recommend reading the whole paper) the award-winning paper "Touring the Calculus Gallery" by William Dunham. We discussed the first half of the paper in a session on The Classical Wing on Saturday November 26th.

    You can download a PDF of Dunham's paper at http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/dunham1.pdf

    Please download and try to work out the 9 exercises I compiled to help guide your reading and our discussion at http://files.meetup.com/3948532/CalculusGallery.Dec2016.pdf. Fully answering these questions is probably too challenging, but the more aspects of each one that you understand before our discussion the easier it will be for you to learn the rest during the event. During our discussion, we will discuss your attempts to solve each of these 9 exercises.

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  • CJ F.

    Tomorrow we will discuss William Dunham's paper http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/dunham1.pdf

    To focus the discussion on some highlights, our exercise set http://files.meetup.com/3948532/CalculusGallery.Dec2016.pdf invites you to think more deeply about some of the key points in the paper.

    To address the last question "Compare and contrast the Reimann integral with the Lebesgue integral", I skimmed the treatment of the Lebesgue integral in Wendell Fleming's "Functions of Several Variables".

    I do not really understand this stuff, but here goes ...

    In the Lebesgue theory, the step functions used to estimate the integral are bounded and measureable. The boundedness restriction can be relaxed by taking limits.

    In Riemann's approach the step functions are comprised of the union of half-open n-dimensional intervals (1-dimensional in the case of single-variable calculus).

    Yesterday

    • CJ F.

      The estimates of measure in the Lebesgue theory continues to puzzle me. In Fleming's treatment, we estimate sets from without by open sets and from within by compact sets. A set is measurable if the estimates agree. But why the asymmetry between inner and outer measure?

      Lebesgue's approach feels more topological working with subtler notions of open sets, compact sets, and closed sets to define measure. I wonder if Fleming's approach is good in the way it engages these complications or bad in its failure to give me an integrated picture?

      Fleming writes "[the Riemann integral] begins with subdividing Eⁿ into intervals, on each of which elementary step functions approximating f from above and below must be constant. Thus, the Lebesgue theory subdivides the 1-dimensional range of f, while the Riemann theory subdivides the n-dimensional domain of f."

      The essay by Dunham says essential the same thing.

      Yesterday

    • CJ F.

      I wonder to what extent Lebesgue's theory of measure affects the benefits of his approach (more functions are integrable over more kinds of regions) versus the innovation in subdividing the range of f instead of its domain?

      What are the differences that you see in the Riemann and Lebesgue integrals?

      Yesterday

  • CJ F.

    On Saturday we will discuss William Dunham's paper http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/dunham1.pdf guided by an exercise set http://files.meetup.com/3948532/CalculusGallery.Dec2016.pdf that highlights some key points.

    As Marty so nicely pointed out Cauchy's intermediate value theorem depends on the completeness property for the reals. There are many possible formulations of completeness. Wikipedia gives the intuition: "there are not any “gaps” (in Dedekind's terminology) or “missing points” in the real number line" https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers

    Wendell Fleming's "Functions of Several Variables" suggests that the simplest form is the least upper bound property. He gives it as an axiom: "Any set S of real numbers that has an upper bound has a least upper bound." The least upper bound is called the supremum (sup S). By considering the set T={x:-xϵS}, the infinum or greatest lower bound of S is - sup T (so inf S = - sup T).

    2 days ago

    • CJ F.

      Fleming's approach to the intermediate value theorem is interesting. First he shows that connectedness of sets of real numbers is characterized by being an interval. The proof is not trivial, so I'll omit it. But I'll bring Fleming on Saturday for anyone who wants to see it.

      "From the intuitive point of view, a set should be regarded as connected if it consists of one piece. Thus an interval on the real line E¹ is connected, while the set [0,1]∪[2,3] is disconnected. For more complicated sets, intuition is not a reliable guide." But his technical definition would take us too far afield.

      "A nonempty set J⊂E¹ is an interval if for every x,yϵJ, x<y, the set [x,y] is contained in J." I'm not quite clear what that means either.

      There are 10 kinds of intervals: [a,b]={x:a≤x≤b}, (a,b)={x:a<x<b}, [a,b)={x:a≤x<b}, (a,b]={x:a<x≤b}, [a,∞)={x:a≤x}, (a,∞)={x:a<x}, (-∞,b]={x:b≥x}, (-∞,b)={x:b>x}, E¹=(-∞,∞), and any real number as a singleton.

      2 days ago

    • CJ F.

      Then Fleming proves that for any continuous real-valued function (single variable) f that is continuous on a connected set S, the image of S under f f(S) is connected. And the intermediate value theorem appears as a trivial corollary: "If S is a connected set and f is real-valued and continuous on S, then f(S) is an interval."

      I like this statement: by seeing the intermediate value theorem as a property of a connected set of real numbers (i.e., as an interval) mapped into an image that must also be a connected set of real numbers (i.e., an interval), we reinforce the idea that continuous real-valued functions preserve both connectedness and intervals.

      To my mind that is much more conceptual that Cauchy's statement or the alternative form given in Marty's resource (http://www-history.mc...­).

      How do you conceptualize the intermediate value theorem?

      2 days ago

  • Martin C.

    As Dunham points out, Cauchy's intermediate value proof is incomplete (no pun intended) and requires use of the completeness axiom for real numbers. The proof on this link is another approach, which does not require setting up intervals, and is the way I was taught the theorem. http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/L20.html

    1 · 2 days ago

  • CJ F.

    For Saturday's discussion, the 9 exercises http://files.meetup.com/3948532/CalculusGallery.Dec2016.pdf follow William Dunham's paper http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/dunham1.pdf essentially linearly.

    But there is no need to discuss the exercises linearly. So I'm going skip ahead to exercise 5 on the Baire Category Theorem. I think Marion said she knows something about it.

    The paper assumes some knowledge of density and completeness. The subtleties and language differ a bit between the paper, Wikipedia, and my analysis text by Wendell Fleming. So I'm a bit lost. He's what I understand so far:

    Dunham's quoting of Baire: "a first-category set cannot exhaust an open interval" where a first category set is one where each point belongs to at least one of "a denumberable infinity of nowhere desnse sets P₁, P₂, P₃, ...".

    Wikipedia's coverage references the topology of complete metric spaces: https://en.wikipedia.org/wiki/Baire_category_theorem

    3 days ago

    • CJ F.

      The Dunham paper says "Baire began by defining a set P of real numbers to be nowhere dense if every open interval (α,β) has an open subinterval (a,b) contining no point of P." But what is a dense set? In my Fleming text, he writes "A set A is called dense in B if every point of B is an accumulation point of A". He explains "A point x₀ is an accumulation point of A if every neighborhood of x₀ contains an infinite number of points of A." So A is dense in B if every neighborhood of every point in B contain an infinite number of points of A. I almost understand this. Is it clear to you?

      But is Fleming's set A, Baire's set P? And is Baire taking the set of real numbers to be Fleming's B? Wikipedia gives a number of definitions of density that are not immediately equivalent to my comprehension: https://en.wikipedia.o...­

      3 days ago

    • CJ F.

      Especially if Marion (or anyone else) can help shed some light on this morass, I'd love to discuss it Saturday. Otherwise, I'm feeling we will need a better resource than Dunham's paper to dig into it more deeply. But I'm grateful to learn more about how the weirdness of Cantor's set theory plus Baire's contribution helps to clarify that the derivative of a function must be continuous on a dense subset of the interval on which we care to examine it.

      3 days ago

  • CJ F.

    On Saturday we will discuss the William Dunham paper "Touring the Calculus Gallery": http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/dunham1.pdf

    We will focus on the second half, pages 10-19, sections 7-10. To help guide the discussion I wrote out 9 exercises: http://files.meetup.com/3948532/CalculusGallery.Dec2016.pdf

    Our tour of the calculus will necessarily be cursory. Witness: the content of this paper samples from the material covered in my two semester Real Analysis I & II courses in college which involved three semesters of calculus as prerequisites!!!

    There is no way we can do justice to this broad subject in a 2-3 hour discussion!

    Our approach will be to discuss each participant's understanding and their questions about the content of the paper. Hopefully each of us will be able to deepen our understanding of several aspects of the calculus. Though many questions will have to go unanswered.

    4 days ago

    • CJ F.

      In Fleming a transformation is a vector-valued function taking a domain D ⊂ Eⁿ (Euclidean n-space) into its image f(D) in E^m (Euclidean m-space): 'the word "transformation"­ is supposed to have a geometric flavor which aids intuition.'

      Fleming defines a neighborhood of a point x0 in Eⁿ to be a spherical n-ball centered at x0 with a positive radius δ. He emphasizes the technical details: "the spherical n-ball U = {x:|x-x0|<δ}". A puntured neighborhood simply omits the center x0. |x-x0| is the vector norm or distance function instead of the absolute value function more familiar to introductory calculus students.

      So in Fleming's definition the limit y0, its neighborhood V, and the image of U under f f(U), all live in m-dimensional space. Whereas x, x0, the domain D, and the neighborhood U of x0 all live in n-dimensional space.

      4 days ago

    • CJ F.

      The parallelism between the real-valued function limit (the traditional notion) and Fleming's vector-valued form is striking! They are almost the same except Fleming emphasizes a geometrical idea which for me is more conceptual.

      Yet I still have trouble shifting from the conceptual to the technical details without a lot of missteps. Indeed the technical details continue to overwhelm my limited skills in analytical thinking.

      Can a geometrical approach like Fleming's restore to the calculus the losses that Dunham details from Cauchy's approach "Gone were proofs by picture; gone were appeals to intuition"?

      Can we learn to think of limits geometrically and rigorously?

      How do you like to think of the limit idea?

      How would you compare and contrast the strengths and deficiencies of Cauchy's definition of a limit?

      4 days ago

  • Martin C.

    We talked about the formula for the surface area of a cone. It is fairly easy to derive. Consider a cone with base circle radius r and slant height L. Imagine making a slit from a point on the base to the to the vertex. If the cone is now unfolded, you get a circular sector centered at the vertex with radius L and arc length 2 pi r. The fraction of the whole circle taken by the sector is 2 pi r/(2 pi L) and so the area of the sector is 2 pi r/(2 pi L) pi L^2 = pi L r

    1 · November 27

  • Martin C.

    Any discussion of the history of calculus should at least mention the work of Abraham Robinson and his hyperreal numbers and non-standard analysis. If you substitute for dx in finding an anti-derivative, you are in effect working with infinitesimals. The hyperreals include infinite and infinitesimal numbers, justifying their use by Newton and Leibniz. As near as I can figure, as an oversimplification, the hyperreals use sequences of real numbers to represent numbers. A real number would be an infinite sequence containing only that number. An infinite number could be represented as {1,2,3,....} and an infintesimal could be{1/2, 1/3, 1/4,...} I looked for references. The following looks promising, but I have not had a chance to read. http://mathforum.org/dr.math/faq/analysis_hyperreals.html

    November 20

    • CJ F.

      I have had my eye on the subject of hyperreals & nonstandard analysis.

      I had identified the Dover book "Infinitesimal Calculus" By James M. Henle, Eugene M. Kleinberg (1979) http://store.doverpub...­ as a possibly interesting resource for reading and possibly a Math Counts discussion.

      Hermoso's introduction looks very nice & accessible. It includes two references I had not considered. The Goldblatt book is freely available: https://archive.org/de...­

      It is graduate level: not for me!

      The edited volume is not widely available and sells for over $250 on Amazon.
      Yikes!

      Robinson's classic treatment is, of course, a treatise, and so not for mere mortals like me :)

      "Short introduction to Nonstandard Analysis" by E. E. Rosinger might be an accessible introduction: http://arxiv.org/abs/...­

      Our topic is a tour through the calculus guided by discussing Dunham's paper. It does not pretend to be complete.

      November 20

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  • CJ F.
    Explorer in Universe, Organizer,
    Event Host

    BA in mathematical sciences from Binghamton University, 1989. I'm working my way... more

  • Sam B.
    Co-Organizer,
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    One day the heavens shined down and offered onto the world SAM: Science, arts and mathematics...­. more

  • Martin C.

    I have a degree in math and work as a computer programmer. I have been looking for... more

  • Marion C.

    PhD in math from Wesleyan. Math research, math book reviews, am also a poet. My book Crossing... more

  • David S.

    MS Courant Institute at NYC, BS Math MIT I have a deep respect and... more

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    hi i'm jeannie

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    I have an Engineering PhD from the University of California, Irvine & professional experience in... more

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