Math 522
Homework 2 Solutions
Page 144
12. Suppose A is not closed, then it does not contain all of its limit points. Let
p0 C (A) be a limit point of A. Then r, B(p0 , r) A = . Thus, there is no r
such that B(p0 , r) C (A) so C (A) is not open.
13. C
K
K
Math 522
Homework 9 Solutions
1. To solve this integral we will use a change of variables to n-dimensional spherical
coordinates and consider the limit as the radius variable goes to innity. The
variables in our new coordinate system will be r, 1 , 2 , .
Math 522
Homework 14 Solutions
Page 492
6. The surface, E is given by
We calculate div v = 2 so
2
2
2
x2
2
4
div v dV =
E
=
=
8
3
4x1
4x1
2 dx3 dx1 dx2
4 4 x1 dx1 dx2
x2
2
2
=
4
2
(4 x2 )3/2 dx2
2
2
128
3
= 16
/2
cos4 d
/2
Now, let S1 = E1 E and consider
Math 522
Homework 12 Solutions
Page 461
4. Let r(s, t) = sin(s) cos(t), sin(s) sin(t), cos(s) and dene D1 and D2 as
D1 = cfw_(s, t) : s [0, /2], t [0, 2]
D2 = cfw_(s, t) : s (/2, ], t [0, 2].
Let S1 be r restricted to D1 and S2 be r restricted to D2 .
Wri