Use stokes' theorem to evaluate s curl f · ds. f(x, y, z) = 5y cos(z) i + ex sin(z) j + xey k, s is the hemisphere x2 + y2 + z2 = 4, z ≥ 0, oriented upward. step 1 stokes' theorem tells us that if c is the boundary curve of a surface s, then curl f · ds s = c f · dr since s is the hemisphere x2 + y2 + z2 = 4, z ≥ 0 oriented upward, then the boundary curve c is the circle in the xy-plane, x2 + y2 = 4 correct: your answer is correct. seenkey 4 , z = 0, oriented in the counterclockwise direction when viewed from above. a vector equation of c is r(t) = 2 correct: your answer is correct. seenkey 2 cos(t) i + 2 correct: your answer is correct. seenkey 2 sin(t) j + 0k with 0 ≤ t ≤ 2π.

By Stokes' theorem, the integral of the curl of across is equal to the integral of along the boundary of , call it . Parameterize by

with . So we have

the infinity sign

step-by-step explanation:

step-by-step explanation:

12: 00 p.m. to 7: 00 p.m. is a total of 7 hours.

4: 00 p.m. to 7: 00 p.m. is a total of 3 hours.

this means that from 4: 00 p.m to 7: 00 p.m. this represents 3/7 of what was lost from 12: 00 p.m. to 7: 00 p.m.

after knowing this, after doing 11,600 - 6,800 = 4,800

4,800 represents 3/7 of what was lost for 7 hours.

we then divide 4,800 by 3 which gets 1,600.

because we know that 1,600 represents 1/7 of how much water was lost, we just have to multiply by 7, which gets 11,200.   after adding 11,200 to how much water was left at 7: 00 p.m. which was 6,800. 11,200 + 6,800 = 18,000

this means that the pool originally had 18,000 gallons in the pool.

you can also check your work by doing 18,000 - 4(1600), which becomes           1800 - 6400 which gets you 11,600.