Name
ID
MATH 253
Section
EXAM 3
Sections 501-503
Spring 1998
8
P. Yasskin
9
Multiple Choice: (5 points each)
1. If F
a.
b.
c.
d.
e.
2. If F
a.
b.
c.
d.
e.
1-7
10
F
Ÿxy, yz, xz then y"zx
Ÿ"y, z, "x
xyz
Ÿ"y, "z, "x
"x y " z
•F
Ÿxy, yz, xz then y"zx
Ÿ"y, z, "x
xyz
Ÿ"y, "z, "x
"x y " z
3. Compute the line integral ; y dx " x dy counterclockwise around the semicircle
x 2 y 2 4 from Ÿ2, 0 to Ÿ"2, 0 .
a. "4=
b. "2=
c. =
d. 2=
e. 4=
(HINT: Parametrize the curve.)
ds for the vector field F
4. Compute the line integral ; F
rŸt e cos
t2
, e sin
t2
for 0 t t t
=.
1 , 1 along the curve
x y
(HINT: Find a potential.)
a. "2
b. 0
c. 2
e
d. 1
e. =
1
5. Compute >Ÿ5x 3y dx Ÿx " 2y dy counterclockwise around the edge of the rectangle
1 t x t 5, 3t y t 6 .
a. 36
b. 24
c. 12
d. "24
e. "36
(HINT: Use Green’s Theorem.)
dS for the vector field F
6. Compute ;; F
C
surface of the solid cylinder C a. 360=
b. 180=
c. 90=
d. 60=
e. 30=
7. If
a.
b.
c.
d.
e.
Ÿx, y, z
zx 3 , zy 3 , z 2 Ÿx 2 y 2
over the complete
3 x 2 y 2 t 4, 0 t z t 3 .
fŸx, y, z x sinŸyz " y cosŸxz z tanŸxy
z sinŸyz z cosŸxz xy sec 2 Ÿxy
sinŸyz " cosŸxz tanŸxy
cosŸyz sinŸxz sec 2 Ÿxy
0
Does not exist.
f •
then 2
8. (25 points) Green’s Theorem states that if R is a nice region in the plane and R is its
boundary curve traversed counterclockwise then
Q
" P dx dy > Pdx Q dy
;;
y
x
R
R
Verify Green’s Theorem if P "y 3 and Q x 3 and R is the region inside the circle
x 2 y 2 9.
Q
8a. (5 points) Compute
" P . (HINT: Use rectangular coordinates.)
x
y
8b. (10 points) Compute ;;
R
Q
" P
y
x
dx dy.
a. (HINT: Switch to polar coordinates and don’t forget the Jacobian.)
8c. (10 points) Compute
>
Pdx Q dy . (HINT: Parametrize of the boundary circle.)
R
3
9. (30 points) Stokes’ Theorem states that if S is a surface in 3-space and S is its
boundary curve traversed counterclockwise as seen from the tip of the normal to S
then
•F
dS > F
ds
;; S
S
Verify Stokes’ Theorem if F Ÿ"yx , xy , x y and S is the part of the cone
z x 2 y 2 below z 2 with normal pointing in and up.
2
2
2
2
•F
. (HINT: Use rectangular coordinates.)
9a. (5 points) Compute •F
dS.
9b. (10 points) Compute ;; S
a. (HINT: Here is the parametrization of the cone and the steps you should use.
Remember to check the orientation of the surface.)
RŸr, 2 Ÿr cos 2, r sin 2, r
r R
2 R
N
•F
;;
Ÿr, 2
R
•F
dS S
4
9c. (15 points) Compute
>
ds .
F
Recall F Ÿ"yx 2 , xy 2 , x 2 y 2 .
S
(HINT: Parametrize of the boundary circle. Remember to check the orientation of
the curve.)
rŸ2 vŸ2 rŸ2
F
>
ds F
S
5
10. (10 points)
The spider web at the right is the graph
of the hyperbolic paraboloid z xy .
It may be parametrized as
Ÿr, 2 Ÿr cos 2, r sin 2, r 2 sin 2 cos 2 .
R
Find the area of the web for r t
8.
6