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integral theorem - Transformation of Double Integrals into...

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Transformation of Double Integrals into Line Integrals - Green’s 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, McGraw-Hill, 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 Green’s (1793-1841) 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
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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 Γ = + 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 Γ = - = - - i (7) From (3) and (7), it becomes evident that P dxdy Pdx y Γ - = ∫∫ i (8) Similarly, using (2), one obtains immediately 2
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( 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 Γ = ∫∫ i (10) Adding (8) and (10) completes the Green’s theorem which reads ( 29 Q P Pdx Qdy dxdy x y Γ + = - ∫∫ 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 Γ = ∫∫ 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 Γ = = - ∫∫ i By adding both areas, one obtains ( 29 1 2 A xdy ydx Γ = - This interesting formula expresses the area of W in terms of a line integral over the boundary.
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