PILOT Problem Set 8.pdf - Math 202 Pilot Problem set 8 week of March 28 1 Let C~1 be given by x = cos y = sin2 0 2 and C~2 be the straight 2 2 ~ line

PILOT Problem Set 8.pdf - Math 202 Pilot Problem set 8 week...

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Math 202 Pilot Problem set 8, week of March 28 1. Let ~ C 1 be given by : x = cos θ, y = sin 2 θ, 0 θ 2 π and ~ C 2 be the straight line segment from (1,0) to (0,1). Let u = x 2 - xy - y 2 and ~ F ( x, y ) = u ( x, y ). Verify directly that R ~ C 1 ~ F · d~s = R ~ C 2 ~ F · d~s . 2. What is the work done by the force field ~ F ( x, y ) = < x 2 - y 2 , - 2 xy > in moving a particle of unit mass counterclockwise around the square bounded by the coordinate axes and the lines x = 3 , y = 3? 3. Let ~ F ( x, y ) = < x r cos r, y r sin r > where r = p x 2 + y 2 . Find R ~ C ~ F · d~s where ~ C is the unit circle centered at the origin traversed counterclock- wise from (1,0) to (0,1). b. where ~ C is the unit circle centered at the origin traversed once around counter- clockwise. c. Is ~ F conservative? 4. Let ~ F = < x - y x 2 + y 2 , x + y x 2 + y 2 > . a. Find R ~ C ~ F · d~s where ~ C is the unit circle centered at the origin traversed once around
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