# Counterclockwise Circulation vs. Clockwise Circulation

1. Dec 17, 2014

### Differentiate1

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. Dec 17, 2014

### LCKurtz

It will be if the $-8\pi$ is correct. Is your polar element of area correct?

3. Dec 18, 2014

### Differentiate1

Yes, I checked and -8π is the correct counterclockwise circulation.
Thanks.

4. Dec 19, 2014

### vela

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