Question

Let S be the part of the surface
z = 16 − (x^2 + y^2)^2
above the xy-plane, oriented upward. Let D be the disk in the xy-plane given by
x^2 + y^2 ≤ 4
, oriented upward. Let
F = yz i  + xz j  + (−11 + xy) k 


(a) Compute divF=



(b) Compute integral _D F · dA =
(c) Compute integral_S F · dA=

Answer 1

here I got the divf = 0
but I dont know how to get the integral _ D F * DA= if the divf is zero, Do i just need the volume of the disk?

Answer 2

@ganeshie8 can you help me please

Answer 3

div(F) = 0
so the flux through any closed surface is 0

Answer 4

for part b,
you want to set up a flux integral through the disk in xy plane

Answer 5

try this
http://www.wolframalpha.com/input/?i=%5Cint_0%5E%7B2pi%7D%5Cint_0%5E2+%28-11%2Br%5E2%5Ccos%5Ctheta%5Csin%5Ctheta%29r+drd%5Ctheta

Answer 6

thanks @ganeshie8 what about part c?

Answer 7

its just algebra, you have below :
total flux out of closed surface = 0
flux out of bottom disk = -44pi

so flux out of top surface = ?

Answer 8

-44pi

Answer 9

it has to be the same flux going in and going out right?

Answer 10

total flux out of closed surface = flux out of disk + flux out of top surface

Answer 11

0 = -44pi + flux out of top surface

Answer 12

flux out of top surface = ?

Answer 13

divergent thm right? :)

Answer 14

weird is marking -44pi right and not 44pi

Answer 15

-44pi for part b
+44pi for part c

Answer 16

for part c is marking -44pi right.. weird

Answer 17

but i guess its because if negative flux its going in negative flux has to come out

Answer 18

not sure, they need to add up to 0