Math 273
Quiz 12
Name:
26 Apr 2016
Show appropriate work.
Point Values in boxes .
Today we are interested in the flux through a silo S. For the purpose of this worksheet, define a silo to
be the cylinder x2 + y 2 = 4 for 0 ≤ z ≤ 6 with a hemispherical cap on the top. There is no bottom on
our silo. Normal vectors point outward.
Figure 1: My grandpa’s farm, complete with silos.
On this page we are interested in using The Divergence Theorem to compute
ZZ
F · dS
S
where F = x + cos y 2 + z 2016 , y + sin x2 + z 2016 , y 2 .
1. Since the Divergence Theorem requires the surface be closed, we start need to do a little work up
front.
(a) 2 Let Sf be the surface representing the floor with downward normal. Compute the surface
integral
ZZ
F · dS.
Sf
(b) 1 Since S + Sf is now a closed surface, use the Divergence Theorem to compute
ZZ
F · dS.
S+Sf
(c) 1 Subtract to evalate
ZZ
F · dS.
S
On this page, we are interested in using Stokes’ Theorem in two different ways to compute
ZZ
curlF · dS.
S
Below are two vector field F. For each, choose an appropriate way to use Stokes’ Theorem and
then do so to evaluate the integral.
D
E
2
2
2. 3 F = xz 2016 − y, x + y + z sin x2016 , ex +y + z 2016
3. 3 F = yz 2 + zx2016 , xy 2 + y 4.26 + zez , x2 y 2 − xyz