Name
ID
MATH 311
Section 200
1.
Exam 3
Spring 2001
P. Yasskin
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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 ™ ; fŸt gŸt dt
˜
0
a.
Ÿ
10 Find cos 2 where 2 is the angle between v 1 and v 2 .
b.
Ÿ
10 Apply the Gram-Schmidt procedure to £v 1 , v 2 ¤ to produce an orthonormal basis £u 1 , u 2 ¤.
1
2.
10 points In R 5 find the volume of the parallepiped with edges
a Ÿ1, 0, 2, 0, 1 b Ÿ0, 2, 1, 0, "1 c Ÿ"1, 0, 0, 2, 1 2
3.
30 points A paraboloid in R 4 with coordinates Ÿw, x, y, z , may be parametrized by
Ÿ
Ÿr, 2 Ÿr 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. Ÿ10 Find the area of the surface.
b.
Ÿ
10 Compute P ;; 1 8w2 8x 2 dS over the surface.
c.
Ÿ
10 Compute I ;;Ÿxy dw dz " 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.
Ÿxz 2 , yz 2 , zŸx 2 y 2 .
Let G
G
.
5 Compute a.
Ÿ b.
Ÿ
G
dV over the solid paraboloid V.
10 Compute ;;; V
HINT: Use cylindrical coordinates.
4
c.
Ÿ
dS over the paraboloid P with normal pointing DOWN and OUT.
15 Compute ;; G
Pw
HINT: Parametrize the paraboloid with coordinates Ÿr, 2 .
d.
dS over the disk D with normal pointing UP.
5 Compute ;; G
Ÿ Du
HINT: Parametrize the disk with coordinates Ÿr, 2 .
e.
dS ;; G
dS ;; G
dS
5 Compute ;; G
Ÿ V
Pw
Du
(Note: By Gauss’ Theorem, the answers to (b) and (e) should be equal.)
5