midterm3review.pdf - MATH 2263 Midterm 3 Review Cora Brown...

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MATH 2263 Midterm 3 Review Cora Brown 1. Sketch the vector field (a) F x, y 1 2 i y x j (b) F x, y y, y 2 2. Evaluate the line integral (since none of these curves are closed, we cannot use Green’s Theorem). (a) C x ds , C is the arc of the parabola y x 2 from 0 , 0 to 1 , 1 (b) C yz cos x ds , C : x t, y 3 cos t , z 3 sin t , 0 t π (c) C xy dx y 2 dy yz dz where C is the line segment from 1 , 0 , 1 to 3 , 4 , 2 (d) C F d r where C is given by r t sin t , 1 t and 0 t π . 3. Show that F is conservative (without finding the potential) and then find the potential function f such that F f . (a) F x, y 1 xy e xy , e y x 2 e xy (b) F x, y, z sin y , x cos y , sin z 4. Show that F x, y 4 x 3 y 2 2 xy 3 , 2 xy 4 3 x 2 y 2 , 4 y 3 is conservative and use that to evaluate C F d r where C is the curve given by r t t sin πt , 2 t cos πt for 0 t 1. 5. Use Green’s theorem to evaluate C xy 2 dx x 2 y dy where C consists of the parabola y x 2 from 1 , 1 to 1 , 1 and the line segment from 1 , 1 to 1 , 1 . Check to make sure C has positive orientation first.
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