# Homework, MATH 411, Spring 2013

Homework is from our text (Fitzpatrick) unless indicated.
Homework is due at the beginning of class on the day listed in the schedule of work.

### Homework #1

10.1:   1,5,6,7 (Hint: Cauchy-Schwarz Inequality),9
10.2:   3, 4 (you could use Theorem 10.9)

### Homework #2

10.3:   1,2 (but in 1,2 you do not need to justify your answers); 10,11
11.1:   1,4,9
Problem A on the page of Extra homework problems.
12.2: 1,3,15

11.2:   1, 6, 10
11.3:   3, 4,6,8
11.4:   1,3, 5

### Homework #4

13.1:   1, 3a, 3c, 9
13.2:   1a, 3, 5, 8a, 11
13.3:   1ab, 7,8,9ab,11

### Homework #5

14.1: 2,3,4,15
14.2:   1,2,9,10
14.3:   1ab, 4, 5, 11d
(Before doing 14.3.11d: compute the polynomial p of degree at most 2 which best approximates f near the origin, where f is the map defined by f(x,y) = cos(x-y+xy) as in 11d. To do this you could plug x-y-xy into enough (not many!) terms of the power series for cos.)

### Homework #6

15.2:   1,3, 8
Problem B from the Extra homework problems
15.3:   1, 9, 11
Problem C from the Extra homework problems

### Homework #7

16.1:  3,5,13,14
(You can "explain analytically" by showing the derivative matrix is not invertible at (x_0,y_0).)
16.2:  3,5, 8
16.3:  1,2,7,11

### Homework #8

17.1:  1,2,3,4,7,12
(For 12, you can use 11 without proving 11.)
17.2:  2,3,5,9

### Homework #9

17.3:  1, 3a, 4 [in 4, look for the simplest linear example]
17.4:   4,7,9,13, 14
Problem D on the page of Extra homework problems.

18.1:   10,14
18.2:   4,8,9
18.3:   5,9,10

### Homework #11

19.1:   2c, 3,5,6,9
[For 19.1: With every iterated integral you use: draw a sketch of the domain of integration, with a line segment indicating the interval of the first (inner) integration, and coordinate axis markings indicating the lower and upper limits of the inner integration.]

### Homework #12

Problem E from the Extra homework problems
19.2:   2, 3,7,8
[Hint on 2: do a linear change of coordinates to reduce to an integral on the unit sphere, and then use spherical coordinates there.]
[Hint on 3: consider change of variables with cylindrical coordinates, which is given by the rule sending (r,theta , z) to (r cos (theta) , r sin (theta), z) . Check an appropriate determinant is nonzero to justify this.]
19.3: 5