PILOT Problem Set 12.pdf - Calculus III PILOT Problem set 12 week of April 25 1 Let S be the part of the parabola zR = 4 x2 y 2 that lies above the R ~

PILOT Problem Set 12.pdf - Calculus III PILOT Problem set...

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Calculus III PILOT Problem set 12, week of April 25 1. Let S be the part of the parabola z = 4 - x 2 - y 2 that lies above the square 0 x 1 , 0 y 1. Calculate R R S ~ F · d ~ S where ~ F = < xy, yz, zx > and S is oriented by the upward normal. 2. Let ~ F = < - y 2 , x, z 2 > and let C be the curve of intersection of the plane y+z=2 with the right circular cylinder x 2 + y 2 = 1. Orient C counterclockwise when viewed from above. Use Stoke’s theorem to evaluate Z C ~ F · d~s . 3. Let ~ F = < y, z, x > and let S be the part of the unit sphere (centered at the origin) in the positive octant oriented outward. Calculate R R S ∇× ~ F · d ~ S . 4. Let ~ V = < 1 , 1 , 1 > and let R be the solid region in the positive octant bounded by the unit sphere (centered at the origin) and the three coordinate planes. Orent the boundary of R by the outward unit normal. Use the di-
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  • Fall '10
  • Khan
  • Calculus, Stokes' theorem, vergence

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