Math 241 Final Exam
Spring 2008
Name
1. Which of the following families of functions are orthogonal on the indicated sets?
Justify your answers.
(a) {sin(nπx):n=1,2, . . . }on the interval [−1
2,1
2].
(b) {e2niπx:n=1,2,3, . . . }on the interval [0,3
4].Remember: for complex
valued functions we use the inner product defined by
hf,gi =
3
4
Z
0
f(x)g(x)dx.(1)
2. Which of the following orthogonal families are complete on the indicated sets?
(a) {sin(2nπx):n=1,2, . . . }on the interval [0,1
2].
(b) {sin(2nπx)sin(mπx):n=1,2,3, . . . ;m=1,2,3...}on the unit square
[0,1] × [0,1].
3. Find a solution to the stationary heat equation
∇2u=0
in a cylinder described as the the region in three-space R3delimited by the surfaces
z=0,z=1,x2+y2=4
assuming that:
•the lower base and the lateral surface are in thermal contact with a thermostat
at temperature u=0
