Name
ID
Section
MATH 253 Honors
EXAM 3
Sections 201-202
1-4
/32
Fall 1999
5
/25
P. Yasskin
6
/25
7
/20
Multiple Choice: (8 points each)
1. Compute the line integral
B
ds
; F
Ÿyz, "xz, z F
of the vector field
A
H
the helix H parametrized by
and
B Ÿ"3, 0, 4= .
a. 20=
b. 26=
c. 26= 2
d. "20=
e. "10= 2
rŸt Ÿ3 cos t, 3 sin t, 4t between
along
A Ÿ3, 0, 0 rŸt Ÿ3 cos t, 3 sin t, 4t 2. Find the total mass of the helix H parametrized by
between
2
A Ÿ3, 0, 0 and
B Ÿ"3, 0, 4= if the linear mass density is
>z .
3
16
=
a.
b.
c.
d.
e.
3
40= 3
3
80= 3
3
16= 3
80= 3
>Ÿ"xy 2 dx x 2 y dy the triangle whose vertices are
Theorem.
a. 4
b. 8
c. 16
d. 32
e. 64
3. Compute
counterclockwise around the complete boundary of
Ÿ0, 0 , Ÿ2, 4 and Ÿ0, 4 .
HINT: Use Green’s
along the curve
;Ÿyz dx xz dy xy dz t
rŸt e
1 =
between t 0 and t =.
Ÿyz, xz, xy .
potential for F
a. " ln 2
b. 1 " ln 2
c. "1 " ln 2
d. 1 ln 2
e. "1 ln 2
4. Compute
sin 4t , cos 5t, ln
HINT: Find a scalar
1
5. (25 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 Ÿy, "x, x 2 y 2 and S is the cone z x 2 y 2 for z t 2
with normal pointing up and in.
The cone may be parametrized by:
Ÿr, 2 Ÿr cos 2, r sin 2, r R
5a. (16 points) Compute
;;
•F
dS
using the following steps:
S
•F
r R
2 R
N
•F
;;
Ÿr, 2 R
•F
dS S
2
and S is the
5b (9 points) Recall
F Ÿy, "x, x 2 y 2 2
2
cone
z x y
with normal pointing up and in.
Compute
using the following steps:
> F ds
S
(Remember to check the orientation of the curve.)
rŸ2 vŸ2 rŸ2 F
ds > F
S
3
6. (25 points) Gauss’ Theorem states that if V is a solid region and V is its boundary
surface with outward normal then
F
dS
dV ;; F
;;; V
V
Verify Gauss’ Theorem if
F Ÿxz, yz, x 2 y 2 and V is the solid hemisphere
0tzt
4 " x2 " y2 .
Notice that V consists of two parts:
the hemisphere H:
z
4 " x2 " y2
x 2 y 2 t 4 with z 0
and a disk D:
6a. (5 pts) Compute
F
dV.
;;; V
HINT: Compute the divergence in rectangular and the integral in spherical.
F
F
dV ;;; V
6b. (8 pts) Compute
dS.
;; F
(HINT: You parametrize the disk.)
D
Ÿr, 2 R
r R
2 R
N
Ÿr, 2 R
F
dS ;; F
D
4
Recall
0tzt
F Ÿxz, yz, x 2 y 2 4 " x2 " y2 .
6c. (9 pts) Compute
dS
;; F
and V is the solid hemisphere
over the hemisphere parametrized by
H
ŸI, 2 Ÿ2 sin I cos 2, 2 sin I sin 2, 2 cos I R
I R
2 R
N
ŸI, 2 R
F
N
F
dS ;; F
H
6d. (3 pts) Combine
dS
;; F
H
and
dS.
;; F
D
to obtain
dS.
;; F
V
Be sure to discuss the orientations of the surfaces (here or above) and give a
formula before you plug in numbers.
dS ;; F
V
5
7.
(20 points) The paraboloid at the right is
the graph of the equation
z 4x 2 4y 2 .
It may be parametrized as
Ÿr, 2 Ÿr cos 2, r sin 2, 4r 2 .
R
Find the area of the paraboloid for z t 16.
6