Counterclockwise Circulation vs. Clockwise Circulation

  • #1

Homework Statement


[/B]
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

Homework Equations



∫∫ ∇xFk dA

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)?
 

Answers and Replies

  • #2
LCKurtz
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Homework Statement


[/B]
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

Homework Equations



∫∫ ∇xFk dA

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)?

It will be if the ##-8\pi## is correct. Is your polar element of area correct?
 
  • #3
Yes, I checked and -8π is the correct counterclockwise circulation.
Thanks.
 
  • #4
vela
Staff Emeritus
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You should recheck the area element again, even though you accidentally got the right answer.
 

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