8 6 points give a curve γ and a differential form ω

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8. [6 points] Give a curve γ and a differential form ω such that integraldisplay γ ω is equivalent by Green’s Theorem to integraldisplay 2 - 2 integraldisplay 3 q 1 - x 2 4 - 3 q 1 - x 2 4 2 xy dy dx . EVALUATION OF THE INTEGRAL IS NOT REQUIRED. 9. [7 points] (a) State the corollary to Green’s Theorem which gives the area of a region as a line integral. (b) Use a line integral to find the area enclosed by the teardrop curve, parametrized by γ ( t ) = ( 2 cos t - sin(2 t ) , 2 sin t ) , 0 t 2 π . x y 10. [10 points] For each of the following differential forms ω determine if ω is exact. If ω is exact, use the algorithm given in class to find its potential function g . (a) ω = ( e yz + 2 xz 2 ) dx + ( x + 1) ze yz dy + ( ( x + 1) ye yz + 2 x 2 z + cos z ) dz (b) ω = yz sin( xyz ) dx + xz sin( xyz ) dy + xyz sin( xyz ) dz
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