Name
ID
MATH 253
Section
EXAM 3
Fall 1998
Sections 501-503
P. Yasskin
Multiple Choice: (7 points each)
1. If
a.
b.
c.
d.
F yz cos x, y sin x, z sin x
"yz sin x
Ÿ2 " yz sin x
"yz sin x, " sin x, sin x
0, Ÿz " y cos x, Ÿy " z cos x
a.
b.
c.
d.
F
then
•F
/49
8
/25
9
/15
10
/25
"yz sin x, sin x, sin x
e.
2. If
then
1-7
F yz cos x, y sin x, z sin x
0, Ÿy " z cos x, Ÿy " z cos x
2Ÿy " z sin x
"yz sin x, " sin x, sin x
0, Ÿz " y cos x, Ÿy " z cos x
e. 0
3. Find a scalar potential for
F yz cos x, y sin x, z sin x .
a. "yz sin x
b. yz sin x
y2
sin x,
2
y2
sin x d. yz sin x 2
e. Does Not Exist
c.
yz sin x,
z 2 sin x
2
z 2 sin x
2
1
4. Compute the line integral
B
ds
; F
of the vector field
A
H
helix H parametrized by
B Ÿ"3, 0, 3= .
a. 2= 2 " 18=
b. 2=
c. 3=
2
d. 9= " 27=
2
2
9
=
e.
2
rŸt Ÿ3 cos 2, 3 sin 2, 2 5. Find the total mass of the helix H parametrized by
between
A Ÿ3, 0, 0 and
B Ÿ"3, 0, 3= > 3 2z.
a. 9Ÿ= = 2 between
Ÿy, "x, z F
along the
A Ÿ3, 0, 0 and
rŸt Ÿ3 cos 2, 3 sin 2, 2 if the linear mass density is
b. 27Ÿ= = 2 c. 9 10 Ÿ= = 2 10 Ÿ6= 4= 2 e. 6= 4= 2
d.
2
>Ÿ4x " 3y dx Ÿ3x " 2y dy
triangle with vertices
Ÿ0, 0 ,
Ÿ0, 3 (HINT: Use Green’s Theorem.)
a. 3
b. 6
c. 9
d. 12
e. 18
6. Compute
7. Compute
•F
dS
;; counterclockwise around the edge of the
and
for the vector field
Ÿ2, 0 .
Ÿ"y, x, z F
over the paraboloid
P
z x2 y2
for
zt9
with normal pointing in and up.
(HINT: Use Stokes’ Theorem.)
a. 2=
b. 3=
c. 4=
d. 9=
e. 18=
3
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
and
Verify Green’s Theorem if
and R is the region inside
Q xy 2
P "x y
2
2
4.
the circle
x y
Q
a. (5 pts) Compute
" P .
(HINT: Use rectangular coordinates.)
y
x
2
b. (10 pts) Compute
;;
R
Q
" P
y
x
dx dy.
(HINT: Switch to polar coordinates and don’t forget the Jacobian.)
c. (10 pts) Compute
> Pdx Q dy .
(HINT: Parametrize the boundary circle.)
R
4
9.
(15 points) The spiral ramp at the right
may be parametrized as
Ÿr, 2 Ÿr cos 2, r sin 2, 2 R
for
0trt2
Compute
and
0 t 2 t 3=.
;; x 2 y 2 dS
over
this spiral ramp.
5
10. (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
and V is the solid region above the
Verify Gauss’ Theorem if
F Ÿxz yz z 2
2
paraboloid
below
the
plane
Notice that V consists of two
z x y
z 4.
parts: a piece of the paraboloid P and a disk D.
2,
a. (7 pts) Compute
F
dV.
;;; 2, 3
(HINTS: Compute
F
in rectangular
V
coordinates. Integrate in cylindrical coordinates.)
F
F
dV ;;; V
b. (8 pts) Compute
dS.
;; F
(HINT: Here is the parametrization of the
P
paraboloid.)
Ÿr, 2 Ÿr cos 2, r sin 2, r 2 R
r R
2 R
N
Ÿr, 2 R
F
dS ;; F
P
6
c. (7 pts) Compute
dS.
;; F
(HINT: You parametrize the disk.)
D
Ÿr, 2 R
r R
2 R
N
R
Ÿr, 2 F
dS ;; F
D
d. (3 pts) Combine
dS
;; F
P
and
dS.
;; F
D
to obtain
dS.
;; F
V
Be sure to discuss the orientations of the surfaces and to give a formula before
you plug in numbers.
dS ;; F
V
7