7 since γ t is a counterclockwise parametrization of

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Unformatted text preview: 7. Since γ ( t ) is a counterclockwise parametrization of a simple closed curve, the area of the contained region can be given by the corollary. Hence, A = 1 2 integraldisplay γ xdy- y dx = 1 2 integraldisplay 2 π bracketleftbig (2 cos t- sin 2 t )(2 cos t )- (2 sin t )(- 2 sin t- 2 cos2 t ) bracketrightbig dt = 1 2 integraldisplay 2 π ( 4 cos 2 t- 2 sin 2 t cos t + 4 sin 2 t + 4 sin t cos 2 t ) dt = 1 2 integraldisplay 2 π parenleftbigg 4- 4 sin t cos 2 t + 4 sin t cos 2 t- a24 a24 a24 a24 a58 =0 2 sin t parenrightbigg dt = 4 π . x y MATB42H Solutions # 5 page 5 8. Here we may calculate directly or we can use the fact that ω =- y x 2 + y 2 dx + x x 2 + y 2 dy is the 1–form used in the definition of the winding number. (a) γ is a circle centered at (1 , 3). Since (0- 1) 2 + (0- 3) 2 = 1 + 9 = 10 > 9, γ does not enclose the origin and its winding number is 0. Hence, integraldisplay γ ω = 2 π ( winding number ) = 2 π (0) = 0.-2 1 4 x 3 6 y (b) Since γ is a closed curve which encircles the origin once in the clockwise direction, its winding number is- 1. Hence, integraldisplay γ ω = 2 π (winding number) = 2 π (- 1) =- 2 π .-0.5 1 x-2 2 y y Plus x Equal Minus 1 y Plus 4x Equal 4 y Minus 2x Equal 1...
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