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Unformatted text preview: Transformation of Double Integrals into Line Integrals  Greens theorem in the plane (Reference, Advanced Engineering Mathematics, Erwin Kreyszig, 1967, Wiley, pg 313 Advanced Engineering Mathematics, Grossman/Derrick, 1988, Harper, pg 611 Mathematic of Physics and Modern Engineering, Sokolnikoff/Redheffer, 1966, McGrawHill, pg 370) Double integrals over a plane region may be transformed into line integrals over the boundary of the region and conversely. This transformation also known as Greens (17931841) theorem is of practical as well as theoretical interest. This concept is extended in a tool called planimeter used to measure the area of a graphically represented planar region. 1 The region W can be represented in both of the form ( 29 ( 29 , a x b u x y v x (1) and ( 29 ( 29 , c y d p y x q y (2) Consider first ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 { } , , , b v x b b a u x a a y v x P P dxdy dy dx P x y dx P x v x P x u x dx y u x y y = = = = = (3) Now 1 2 C C Pdx Pdx Pdx = + i (4) where G is the entire perimeter (closed  counter clockwise). From the sketch, it is immediately clear that ( 29 ( 29 1 , , b C a P x y dx P x u x dx = (5) Similarly, ( 29 ( 29 ( 29 2 , , , a b C b a P x y dx P x v x dx P x v x dx = =  (6) Thus ( 29 ( 29 ( 29 ( 29 { } , , , , b b b a a a Pdx P x u x dx P x v x dx P x v x P x u x dx = =  i (7) From (3) and (7), it becomes evident that P dxdy Pdx y  = i (8) Similarly, using (2), one obtains immediately 2 ( 29 ( 29 ( 29 ( 29 { } , , d q y d d p y c Q Q dxdy dx dy Q q y y Q p y y dy x x = = (9) which by analogous reasoning yields the following equation: Q dxdy Qdy x = i (10) Adding (8) and (10) completes the Greens theorem which reads ( 29 Q P Pdx Qdy dxdy x y + = i (11) Consider the area of a plane region as a line integral over the boundary. In (11), let P=0 and Q=x . Then dxdy xdy = i The left side of the above integral is the area A of W . Similarly, let P=y and Q=0 ; then from (11) A dxdy ydx = =  i By adding both areas, one obtains ( 29 1 2 A xdy ydx = This interesting formula expresses the area of W in terms of a line integral over the boundary....
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 Summer '10
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