18 02A Topic 39 Green s theorem Author Jeremy Orloff Read TB 21 3 to middle p 768 Ingredients C a simple closed curve i e no self intersection R the interior of C C must be positively oriented traversed so interior is on left and piecewise smooth a few corners are okay y R C R R C C x ZZ I Nx My dA M dx N dy Green s Theorem R C If F hM N i then we call Nx My k curlF The reason Z Z is given in I for the name curlF k dA F dr the reading section V 4 We can write Green s theorem as C R In the notes they drop the k from the curl This is confusing since when we move to 3D integrals the curl will have to be a vector Example 1 use the RHS to find the LHS Use Green s Theorem to compute I 3x2 y 2 dx 2x2 1 xy dy where C is the circle shown I y C ZZ ZZ C By Green s Theorem I 6x2 y 4x 6x2 y dA 4 x dA R a ZZ R x 1 We could compute this directly but we know xcm x dA a A R ZZ x dA a3 I 4 a3 R Path for example 1 Example 2 Use the LHS to find the RHS Use Green s Theorem to find the area under one arch of the cycloid x a sin y a 1 cos The picture shows theIcurve C C Z1Z C2 surrounding the area By Green s Theorem y dx dA area C R I Z Z area y dx 0 dx y dx C1 C2 C1 C2 Z 2 a2 1 cos 2 d 3 a2 0 1 2a y C2 C1 a Path for example 2 2 a x 18 02A topic 39 2 Other ways area I ZZ I to compute I 1 dA y dx x dy y dx x dy 2 C R C C Proof of Green s Theorem also see the reading i First we ll work on a rectangle Later we ll use a lot of rectangles to approximate an arbitrary region I I M dx ii We ll only do C M dx y N dy is similar C By direct calculation the right hand side of Green s Theorem ZZ Z bZ d M M dA dy dx y y c R a Inner integral M x y dc M x d M x c ZZ Z b M Outer integral M x c M x d dx dA y R a For I the LHS weZ have Z C d c a M dx M dx since dx 0 along the sides Z a Z b M x c dx M x d dx M x c M x d dx bottom Z b a top b a So for a rectangle we have proved Green s Thm by showing the two sides are the same For line integrals when adding two rectangles with a common edge the common edges are traversed in opposite directions so the sum is just the line integral over the outside boundary Similarly when adding a lot of rectangles everything cancels except the outside boundary This extends Green s Theorem on a rectangle to Green s Theorem on a sum of rectangles Since any region can be approximated as close as we want by a sum of rectangles Green s Theorem must hold on arbitrary regions End of topic 39 notes x b
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