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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