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MIT18_024s11_ChEnotes

MIT18_024s11_ChEnotes - GREEN'S THEOREN AND I T S...

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GREEN'S THEOREN AND ITS APPLICATIONS The discussion i n 11.19 - 11.27 of, Apostol is not complete nor e n t i r e l y rigorous, a s the author himself points out. W e give n here a rigorous treatment. Green's Theorem -- i n t h e Plane We already know what is meant by saying t h a t a region i n t h e plane is of Type . I o r of Type 11 o r t h a t it is of both types simultaneous l y Aposiso 1 proves Green ' s Theorem f o r a region t h a t is of both types. Such a region R can be described i n two d i f f e r e n t ways, a s follows: The author's proof is complete and rigorous except f o r one gap, which a r i s e s from h i s use of t h e i n t u i t i v e notion of "counter- clockwisen. 1
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S p e c i f i c a l l y , what h e does is t h e following: For t h e first: p a r t of t h e proof h e o r i e n t s t h e boundary C of R as follows: ( * ) By i n c r e a s i n g x , on t h e curve y = @I (x) ; By i n c r e a s i n g y , on t h e l i n e segment x = b; By decreasing x, on t h e curve y = $ 2 (x) : and By decreasing y , on t h e l i n e segment x = a. Then i n t h e second p a r t of t h e proof, he o r i e n t s C a s follows: (**) By decreasing y , on t h e curve x = q L ( y ) ; By i n c r e a s i n g x, on t h e l i n e segment y = c; By i n c r e a s i n g y , on t h e curve x = I J ~ ( Y ) ;and By decreasing x, on t h e l i n e segment y = d. (The latter l i n e segment collapses t o a s i n g l e p o i n t i n t h e pre- c e d i ng figure.) The c r u c i a l question is: ---- How does one know t h e s e - two o r i e n t a t i o n s - are - t h e - s ante? One can i n f a c t see t h a t t h e s e two o r i e n t a t i o n s a r e t h e , same, by simply analyzing a b i t more c a r e f u l l y what one means by a region o f ~ ~ I and ~ 11. e s S p e c i f i c a l l y , such a region can be described by four monotonic functions :
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- I / A.. wt..ere dl and d4 are strictly decreasing and X2 and d3 are U w * stzictly increasing, , [some o r a l l of t h e ai can be missing, of 1 course. Here a r e pictures of t y p i c a l such regions:] T h e curves a1 and a2, along with t h e l i n e segment y = c, are used t o define t h e curve y = +1(~) t h a t bounds the region on t h e bottom. Similarly, a3 and a4 and y = d define t h e curve y = C $ ~ ( X ) t h a t bounds t h e region on t h e top.
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Similarly, the inverse functions t o a1 and a3 , along with x = a , combine t o define the curve x = Ql ( y ) t h a t bounds the region on the l e f t ; and the inverse functions t o ag and a d , along with x = b, define the curve x = P2 (y) . NOW one can choose a direction on the bounding curve C by simply directing each of these eight curves a s indicated i n the figure, and check t h a t t h i s is t h e same a s the directions specified i n (*) and (**I . [~ormally, one d i r e c t s these curves - a s follows: increasing x = decreasing y on y = al (x) increasing x o n y = c increasing x = increasing y on y = a2(x) increasing y o n x = b decreasing x = increasing y on y = cr4 (x) decreasing x o n y = d decreasing x = decreasing y on y = a g ( x ) decreasing y tJe make t h e following definition: Definition. L e t R be an open set i n t h e plane bounded by a
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