# Name ID 1 /20 3 ```Name
ID
MATH 311
Section 200
1.
Exam 3
Spring 2001
P. Yasskin
1
/20 3
/30
2
/10 4
/40
20 points Let V be the vector space of functions spanned by v 1 t and v 2 t 2 with the inner
product
1
f, g  ; ft gt dt

0
a.

10 Find cos 2 where 2 is the angle between v 1 and v 2 .
b.

10 Apply the Gram-Schmidt procedure to &pound;v 1 , v 2 &curren; to produce an orthonormal basis &pound;u 1 , u 2 &curren;.
1
2.
10 points In R 5 find the volume of the parallepiped with edges
a 1, 0, 2, 0, 1 b 0, 2, 1, 0, &quot;1 c &quot;1, 0, 0, 2, 1 2
3.
30 points A paraboloid in R 4 with coordinates w, x, y, z , may be parametrized by

r, 2 r cos 2, r sin 2, r 2 , r 2 w, x, y, z R
for 0 t r t 3 and 0 t 2 t 2=.
a. 10 Find the area of the surface.
b.

10 Compute P ;; 1 8w2 8x 2 dS over the surface.
c.

10 Compute I ;;xy dw dz &quot; wz dx dy over the surface.
w r cos 2,
x r sin 2,
y r2,
z r2
3
4.
(40 points) The solid paraboloid V at the right
is given by
x 2 y 2 t z t 4.
It’s boundary (denoted by V) has two parts:
The paraboloid P given by z x 2 y 2 for z t 4.
The disk D given by x 2 y 2 t 4 with z 4.
xz 2 , yz 2 , zx 2 y 2 .
Let G
G
.
5 Compute a.
 b.

G
dV over the solid paraboloid V.
10 Compute ;;; V
HINT: Use cylindrical coordinates.
4
c.

dS over the paraboloid P with normal pointing DOWN and OUT.
15 Compute ;; G
Pw
HINT: Parametrize the paraboloid with coordinates r, 2 .
d.
dS over the disk D with normal pointing UP.
5 Compute ;; G
 Du
HINT: Parametrize the disk with coordinates r, 2 .
e.
dS ;; G
dS ;; G
dS
5 Compute ;; G
 V
Pw
Du
(Note: By Gauss’ Theorem, the answers to (b) and (e) should be equal.)
5
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