Name
ID
Math 311
Exam 3
Spring 2002
Section 503
1-7
P. Yasskin
Multiple Choice (8 points each.)
Circle 3 to grade for part credit:
1
2
8
3
4
5
6
/56
9
/20 10 EC
/24
/10
7
∇ ⋅ F⃗ =
1. If F = xy, yz, xz then ⃗
a. y − z + x
b. −y, z, −x
c. x + y + z
d. −y, −z, −x
e. −x + y − z
2. If F = xy, yz, xz then ⃗
∇ × F⃗ =
a. y − z + x
b. −y, z, −x
c. x + y + z
d. −y, −z, −x
e. −x + y − z
3. If fx, y, z = x sinyz  − y cosxz  + z tanxy  then ⃗
∇ × ⃗∇f =
a. z sinyz  i + z cosxz  j + xy sec 2 xy  k
b. sinyz  i
− cosxz  j + tanxy  k
c. cosyz  i + sinxz  j + sec 2 xy  k
d. 0
e. Does not exist.
4. Compute the line integral ∫ y dx − x dy counterclockwise around the semicircle x 2 + y 2 = 4 from
to −2, 0 .
(HINT: Parametrize the curve.)
2, 0 
a. −4π
b. −2π
c. π
d. 2π
e. 4π
1
⃗ ⋅ ds⃗ for the vector field F
⃗=
5. Compute the line integral ∫ F
⃗rt = e cost 2  , e sint 2 
a.
b.
c.
d.
e.
for 0
≤t≤ π.
1, 1
along the curve
x y
(HINT: Find a potential f so that F = ⃗
∇f.)
−2
0
2
e
1
π
6. Compute ∮5x + 3y  dx + x − 2y  dy counterclockwise around the edge of the rectangle 1
(HINT: Use Green’s Theorem.)
3≤y≤6.
≤ x ≤ 5,
a. 36
b. 24
c. 12
d. −24
e. −36
⃗ ⋅ dS⃗ for the vector field F
⃗=
7. Compute ∫∫ F
∂C
cylinder C =
a.
b.
c.
d.
e.
x, y, z 
zx 3 , zy 3 , z 2 x 2 + y 2 
∣ x 2 + y 2 ≤ 4, 0 ≤ z ≤ 3
over the total surface of the solid
with outward normal. (HINT: Use Gauss’ Theorem.)
360π
180π
90π
60π
30π
2
8.
(20 points) Compute ∫∫ x 2 y dx dy over the
y
20
R
diamond shaped region R bounded by
y = 1x ,
y = 2x ,
y = 22 ,
y = 42
x
x
For full credit you must use curvilinear coordinates.
Half credit for rectangular coordinates.
10
0
1.25
2.5
3.75
x
3
9. (24 points) Stokes’ Theorem states that if S is a nice surface in R 3 and ∂S is its boundary curve traversed
⃗ is a nice vector field on S then
counterclockwise as seen from the tip of the normal to S and F
⃗ ⋅ dS⃗ = ∮ F
⃗ ⋅ ds⃗
∫∫ ⃗∇ × F
S
∂S
Verify Stokes’ Theorem if F = −yx 2 , xy 2 , x 2 + y 2  and S is the part of the cone z =
z = 2 with normal pointing up and in.
x2 + y2
below
9a. (4 points) Compute ⃗
∇ × F⃗. (HINT: Use rectangular coordinates.)
9b. (10 points) Compute
∫∫
⃗∇ × F
⃗ ⋅ dS⃗.
S
(HINT: Here is the parametrization of the cone and the steps you should use. Remember to check the orientation
of the surface.)
⃗ r, θ = r cos θ, r sin θ, r
R
⃗r =
R
⃗θ =
R
⃗=
N
⃗∇ × F
⃗
⃗∇ × F
⃗
∫∫
⃗ r, θ
R
⋅
=
⃗=
N
⃗∇ × F
⃗ ⋅ dS⃗ =
S
4
⃗ ⋅ ds⃗ .
9c. (10 points) Compute ∮ F
Recall F =

−yx 2 , xy 2 , x 2 + y 2  .
∂S
(HINT: Parametrize of the boundary circle. Remember to check the orientation of the curve.)
(CHECK the answers to 9b and 9c agree.)
⃗rθ  =
⃗vθ  =
⃗⃗rθ  =
F
⃗ ⋅ ⃗v =
F
⃗ ⋅ ds⃗ =
∮ F
∂S
5
10. (10 points Extra Credit) Compute the line integral ∮
boundary of the plus sign shown below.
Be sure to justify any theorem you use.
y
x + y2
2
dx −
x
dy counterclockwise around the
x2 + y2
(Hint: The answer is not zero.)
2
1.75
1.5
1.25
1
0.75
0.5
0.25
0
-2-1.75
-1.5
-1.25
-1-0.75
-0.5
-0.25
00.25
0.5
0.75
1 1.25
1.5
1.75
2
-0.25
-0.5
-0.75
-1
-1.25
-1.5
-1.75
-2
6