Name
ID
MATH 253 Honors
EXAM 3
Sections 201-203
Section
Spring 1999
P. Yasskin
Multiple Choice: (5 points each)
1-6
/30
7
/20
8
/30
9
/20
F
1. If F Ÿxz, yx, zy then a.
b.
c.
d.
e.
z"xy
Ÿz, "x, y xyz
Ÿz, x, y "x y " z
•F
2. If F Ÿxz, yx, zy then a.
b.
c.
d.
e.
z"xy
Ÿz, "x, y xyz
Ÿz, x, y "x y " z
y cos x
•F
3. If F Ÿ x2 sin z2 , 2
, tanŸzy then 2
x y
x y
a. sin z " cos2x 2 secŸzy tanŸzy Ÿx
2
Ÿx
2
Ÿx
2
Ÿx
2
y2 b. sin z " cos2x 2 secŸzy tanŸzy y2 c. sin z " cos3x 2y secŸzy tanŸzy sec 2 Ÿzy y2 d. cos z " sin3x 2y secŸzy tanŸzy sec 2 Ÿzy y2 e. None of These
1
4. Compute the line integral ; y dx " x dy counterclockwise around the semicircle
x 2 y 2 9 from Ÿ3, 0 to Ÿ"3, 0 .
a.
b.
c.
d.
e.
(HINT: Parametrize the curve.)
"9=
"2=
=
2=
9=
ds for the vector field F
5. 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. =
dS for the vector field F
6. Compute ;; F
C
surface of the solid cylinder C outward.
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
with normal pointing
360=
180=
90=1
60=
30=
2
y
dx " 2 x 2 dy counterclockwise
x y
x y2
around the boundary of the plus sign shown below.
Be sure to justify any theorem you use.
(Hint: The answer is not zero.)
7. (20 points) Compute the line integral >
2
2
1
-2
-1
0
1
x
2
-1
-2
3
8. (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 Ÿyx , "xy , x z y z and S is the hemisphere
z 4 " x 2 " y 2 with normal pointing up and out.
2
8a. (10 points) Compute
>
2
2
2
ds using the following steps: (Remember to check the
F
S
orientation of the curve.)
rŸ2 vŸ2 rŸ2 F
>
ds F
S
•F
. (HINT: Use rectangular coordinates.)
8b. (5 points) Compute •F
4
•F
dS using the following steps:
8c. (15 points) Compute ;; S
Recall F Ÿyx 2 , "xy 2 , x 2 z y 2 z and S is the hemisphere z normal pointing up and out.
4 " x 2 " y 2 with
Ÿ2, C R
2 R
C R
N
•F
;;
Ÿ2, C R
•F
dS S
5
9.
(20 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