quiz_10D - D or on C .) But x x x 2 + y 2 + y y x 2 + y 2 =...

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Math 213 — Quiz 10 Name Solution 1. Let F = - y i + x j x 2 + y 2 . Show that Z C F · d r = 0 for any simple closed path that does not pass through or enclose the origin. Let D be the region enclosed by C . Then Green’s Theorem states that Z C F · dr = ZZ D ± ∂x ² x x 2 + y 2 ³ + ∂y ² y x 2 + y 2 ³´ dA since the components of F are continuously differentiable on D . (Indeed, the only bad point is the origin, which is not inside
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Unformatted text preview: D or on C .) But x x x 2 + y 2 + y y x 2 + y 2 = 1 x 2 + y 2-2 x 2 ( x 2 + y 2 ) 2 + 1 x 2 + y 2-2 y 2 ( x 2 + y 2 ) 2 = 2 x 2 + y 2-2 x 2 + 2 y 2 ( x 2 + y 2 ) 2 = 0 . Therefore, Z C F dr = ZZ D dA = 0 as claimed....
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