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Math Help - Proof of Green's theorem in Apostol's book

  1. #1
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    Proof of Green's theorem in Apostol's book


    Hello, guys.
    I am working with Apostol's Mathematical Analysis, first edition, Ed. Addison Wesley. My difficult is with a point in theorem 10-42 (p. 287-289), immediately previous (and essential) to Green's Theorem. Please, if you do not have time for helping me, just tell me where I can look for assistance.
    (Sorry for the clumsy redaction, I am not the better in english).
    Well, Apostol has a contour denoted by the greek letter "gamma", described by a function "alpha", which has its component functions, "alpha 1" and "alpha 2", all of them defined in [a,b] as parametric interval.
    At certain point (which is irrelevant for the purpose of my question) Apostol crosses horizontally the contour with a line segment L. This of course produces two new contours, "gamma one" and "gamma two". Be aware that each of these new contours is NOT only an arc of the mother-contour "gamma", because they have a line side that the mother-contour does not: it is L.
    In p. 289 Apostol puts this equation:
    Total variation of real function "alpha 2" on the parametric interval of gamma (which as I said is [a, b]) =
    Total variation or real function "alpha 2" in the parametric interval of gamma 1, plus
    Total variation or real function "alpha 2" in the parametric interval of gamma 2."
    This is my problem: I can not find a way to justify this equation. (For Apostol it may be obvious, because he doesnt prove it). It is not easy to find the parametric intervals corresponding to contours gamma 1 and gamma 2. Remember that these two are not merely arcs of "mother-contour" gamma, because gamma 1 and gamma 2 include also a line segment. For that reason I can not simply take a number from [a, b] (say "c"), state that [a, c] corresponds to contour gamma 1, [c. b] to contour gamma 2, and apply theorem 8.11 (sum of total variations).
    What I am trying to do? Well, I am parametrizing contour gamma 1 this way: a function for its curvilinear section (which is part of contour gamma), another function for its line segment section, and then I concatenate (join) both of them in a single parametrization. I am doing the same for contour gamma 2.
    But I have doubts about this tactic because I have to assume that certain points in the mother contour gamma corresponds, each one of them, to several numbers in the domain of function alpha. An undesirable restriction.
    Please, if you have not enough time to answer, just tell me where I can find assistance. I am studying analysis by myself (no college, no university). I am blocked in this.
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  2. #2
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    hi, I have just read Apostol' Mathematical Analysis months ago, but the second edition. The Green's Theorem does not appear in the second edition, maybe because the author thought these knowledges (Stokes' type of theorems or vector integral calculus) do not belong to entrance mathematical analysis, or not satisfied with his exposition (just guess:-)). I recommend the following two books for Green's Theorem: 1) Spivak's "Calculus on Manifolds", 2)Munkres' "Analysis on Manifolds", if you need to study it in depth for your major, or just skip it if your specialty don't need it at all.
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  3. #3
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    Do you mean that the line integral around \gamma equals the line integral around \gamma_1 and \gamma_2 and you want to see why?

    edit: nevermind
    Last edited by maddas; April 26th 2010 at 07:55 PM.
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  4. #4
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    No, in the proof Apostol does not speak about line integrals. He has a contour (denoted by greek letter gamma) describe by a function alpha, conformed by component functions alpha 1 and alpha 2. Let's suppose the contour gamma is a circle. We write the horizontal diameter. The upper semicircle (including the diameter) is denoted contour gamma 1. The other one (also with the diameter) is denoted gamma 2.

    He says, let's consider: (i) the total variation of alpha 2 on the parametric interval of the circle; (ii) the total variation of the same function on the parametric interval of the upper closed semicircle; and iii) the total variation of the same function on the parametric interval of the other closed semicircle; and he says that (i) = (ii) + (iii).

    I know there is a theorem that goes like this (8.11 in Apostol's book): if we have a < c < b, then the total variation of f on interval (a, b) is equal to the sum of the total variation of f in (a, c) and the total variation of in (c, b).

    But in my case I only know the parametric interval of the circle (it is (a,b)), I do not know the parametric intervals of the semicircles (which include, both, the diameter).

    Someone told me that I only have to find the parametric intervals of the closed semicircles by joining, for each case, the function that describes the arc and the function that describes the diameter, and that afterwards "the line segments, as they are traversed in opposite directions, are anulled". But I failed to see how I can anull something here. I am not talking about line integrals over opposite curves...
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  5. #5
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    What does variation mean? For a real-valued function of a real variable, I know it means the vertical component of the graph's arc length.
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  6. #6
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    Total variation of real function f on the closed interval (a, b) is
    sup {  sum (f(t_1) - f(t_i-1)

    It is cumbersome to write without symbols but let's try. You have a real function f define on a closed interval (a, b). You have a partition P of (a, b) = (t_0,... t_n). Consider the sum over i = 1, ..., n of the absolute values of the diferences between f(t_i) and f(t_i-1). The supremum of all those sums (considering all the partitions) is the total variation of f on the closed interval (a, b).
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  7. #7
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    Yes, that's what I'm familiar with. But the variation of \alpha_2 is just the vertical component of the arclength, and obviously adding a horizontal line will not change this. Put another way, if a particle moves along the contour and casts a shadow on the y-axis, the variation of \alpha_2 is the distance traveled by this shadow. Since as you move along a horizontal line your shadow on the y-axis does not move, the horizontal line makes no contribution to the varaition of \alpha_2.
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  8. #8
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    Thumbs up

    And by that reason the variation of alpha 2 in the circle is equal to the sum of the variations in the two closed semicircles... now I got the idea. I suppose that knowing the parametric intervals of the semicircular contours is not so important once you got the idea... I thank you, maddas.
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