Print Name________________________________
8 pts.
1. A vector function F with continuous partial derivatives on all of
and only if: (circle all that make a true statement)
R 3 , is conservative if
a) F is independent of path.
b) F =
f
for some scalar function f.
c) div(F) = 0
d) curl(F) = 0
j is applied along the
14 pts. 2. Find the work done if the force F xe i xe
path consisting of the line segment from (1,0) to (1,1), followed by the curve along
y
y x 2 from (1,1) to (0,0).
xy
3. 12 pts. A wire in the shape of the curve
density function
2 52
x
5
from x = 0 to x = 2 has
( x , y ) x 2 . Find the mass of the wire.
4. 12 pts.
y
F ( x , y ) (sin( xy ) xy cos( xy )) i x cos( xy ) j
2
a) Find a potential function for
b) Find the work done if
F.
F is applied along any path from ( 2, )
4
to (
1, 0).
5.
14 pts.
F xy i xy j
followed by the path along
the line segment from
and C is the line segment from (0,0) to (1,0)
y 1 x2
from (1,0) to
1 1
,
to (0,0).
2
2
a) Use Green's Theorem to evaluate
F dr
C
b) Use Green's Theorem to evaluate
F n ds
C
1 1
,
2 2
followed by
6.
20 pts. Evaluate
xy
dS where S is the part of the paraboloid
2
2
S x y
z 9 x 2 y 2 above the region bounded by bounded by
x 0, y x , and y 2 x 2 .
y 0, x 0, and y 4 x 2 .
C is the intersection of this cylinder and the paraboloid z x 2 y 2 .
7. 20 pts. Consider the cylinder bounded by
F ( x , y , z ) xz i y j yz k
Use Stokes' Theorem to evaluate
Use the upward normal for S.
C F dr
where C is traversed once.
Bonus question: 4 pts. For the vector field
surface of the hemisphere
F 2z i j 2 y k
x 2 y 2 z 2 25
oriented outw ard. Find the flux of F across S .
and S is the
in front of the plane
y
4
x
3
0
