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