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Counterclockwise Circulation vs. Clockwise Circulation

  1. Dec 17, 2014 #1
    1. The problem statement, all variables and given/known data

    Consider F = (y - z)i + (z - x)j + (x + z)k
    Find circulation of F in the clockwise direction as seen from above, around the boundary of surface S defined by

    z = 4 - x2 - y2
    0 ≤ z ≤ 4

    2. Relevant equations

    ∫∫ ∇xFk dA

    3. The attempt at a solution

    ∇xF = i - 2j - 2k (after solving the cross product)
    The curl of F dot k = -2

    0 ≤ r ≤ 2
    0 ≤ θ ≤ 2π

    ∫∫ -2 drdθ = ∫ -4 dθ = -8π

    -8π is the counterclockwise circulation of F, but since the problem asks for the clockwise direction, would the answer just be 8π (opposite sign)?
     
  2. jcsd
  3. Dec 17, 2014 #2

    LCKurtz

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    It will be if the ##-8\pi## is correct. Is your polar element of area correct?
     
  4. Dec 18, 2014 #3
    Yes, I checked and -8π is the correct counterclockwise circulation.
    Thanks.
     
  5. Dec 19, 2014 #4

    vela

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    You should recheck the area element again, even though you accidentally got the right answer.
     
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