MATH 251 Fall 2023 Section 16 4 Green s Theorem Recall that given a curve C r t hx t y t i for a t b ZC P x y dx Q x y dy Z b a P x t y t x0 t Q x t y t y0 t dt However sometimes the integral on the right can be cumbersome In this section you will learn how to compute the line integral RC P x y dx Q x y dy using a double integral This is called Green s Theorem Note that if F x y P x y i Q x y j then ZC P x y dx Q x y dy Z b Z b ZC a P x t y t x0 t Q x t y t y0 t dt a hP x t y t Q x t y t i h x0 t y0 t idt F dr De nition A closed curve is a curve in which its terminal point coincides with its initial point A simple closed curve is a closed curve that does not cross itself anywhere between its endpoints When integrating over a closed curve C we use the notationHC instead ofRC Green s Theorem Let C be a positively oriented counter closewise piecewise smooth simple closed curve in the plane and let D be the region bounded by C If P and Q have continuous partial derivatives on an open region that contains D then IC P dx Qdy x D Q x P y dA Question If F x y hP x y Q x y i then as above HC P dx Qdy HC F dr But from Section 16 3 isn t it true thatHC F dr 0 If so why do we need Green s theorem to help computeHC P dx Qdy 1 2 Example 1 EvaluateHC x5dx xy2dy where C is the triangular curve consisting of the line segments from 0 0 to 1 0 from 1 0 to 1 1 and from 1 1 to 0 0 Example 2 Evaluate IC 1 3 y3 3x2 dx 1 3 x3 qsin2 y ey dy where C is the positively oriented curve that encloses the region bounded by the lower half of the unit circle x2 y2 1 and y 0 Example 3 Compute the work of the force eld F x y hx4 2y 7x ln y2 1 i on an object that moves along a positively oriented boundary curve of a region with area 12 3 Example 4 Compute the work of the force eld F x y h y ex xy arctan yi on an object moving along the ellipse 1 clockwise x2 9 y2 16
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