Name
MATH 311
ID
Exam 3
Fall 2001
Section 200
1.
35 points
1
/35
2
/35
3
/30
P. Yasskin
⃗ ⋅ dS⃗ over the complete surface of the box
Compute ∫∫ F
∂C
0≤x≤2
⃗ = x 2 y 2 z 3 , xy 3 z 3 , xy 2 z 4 .
where F
0≤y≤3
0≤z≤4
1
2.
35 points Consider the cone C given by
z=
x2 + y2
for z ≤ 4
⃗ = −yz, xz, −xy .
and the vector field F
⃗×F
⃗ ⋅ dS⃗ with
We want to compute ∫∫ ∇
C
4
2
0
normal pointing up and into the cone.
⃗×F
⃗.
5 Compute ∇
a.
 
b.
10  Parametrize the cone using cylindrical coordinates r and θ as the parameters
⃗×F
⃗ ⋅ dS⃗.
and give the range of the parameters. Then explicitly compute ∫∫ ∇

C
2
RECALL: C is the cone z = x 2 + y 2
⃗ = −yz, xz, −xy .
the cone and F
c.

for z ≤ 4 with normal pointing up and into
⃗×F
⃗ ⋅ dS⃗. Be sure to name or quote
10  Describe 2 other ways to compute ∫∫ ∇
C
any Theorem you use and discuss the orientation of any curves or surfaces.
i.
ii.
d.

⃗×F
⃗ ⋅ dS⃗ by one of these two methods.
10  Recompute ∫∫ ∇
C
3
3.
30 points
A hypersurface S in R 4 with coordinates w, x, y, z , may be parametrized by

⃗ r, θ, ϕ  = r cos θ, r sin θ, r cos ϕ, r sin ϕ 
w, x, y, z  = R
for 0 ≤ r ≤ 3, 0 ≤ θ ≤ 2π and 0 ≤ ϕ ≤ 2π.
a.

15  Find the tangent vectors, the normal vector and length of the normal vector.
⃗r =
R
⃗θ =
R
⃗ϕ =
R
⃗ =
N
⃗ =
N
b.
5 Find the hyperarea of the hypersurface.
 
A=
4
RECALL: S is the hypersurface parametrized by
⃗ r, θ, ϕ  = r cos θ, r sin θ, r cos ϕ, r sin ϕ 
w, x, y, z  = R
for 0 ≤ r ≤ 3, 0 ≤ θ ≤ 2π and 0 ≤ ϕ ≤ 2π.

c.
5 Compute P = ∫∫∫ 2w2 + 2x 2 dS over the hypersurface.
 
S
d.
5 Compute Q = ∫∫∫wdy dx dz − 5z dwdx dy  over the hypersurface.
 
S
5