Summer 2011, Math 309 C
Quiz 7
1. Suppose h : R2 R is a harmonic function. In polar coordinates you
found that h(1, ) = sin .
a) Is it possible that h is bounded (give a reason)?
b) What is h(0, 0)?
c) What is the maximum value of h over all of R = cfw_(x
Summer 2011, Math 309 C
Quiz 4
1. Find the positive eigenvalues and the corresponding eigenfunctions
associated to the boundary value problem y + y = 0 with y (0) = 0 and
y (L) = 0. Show all work.
1
Summer 2011, Math 309 C
Quiz 5
1. Find the Fourier sine series for the function
f (x) =
1 :0x1
2 :1<x2
You do NOT need to write down the explicit odd extension.
1
Summer 2011, Math 309 C
Quiz 6
1. Consider a wave of length L = 20 that satises the wave equation
2u
u
9 2 = 2 on 0 < x < 20 and for t > 0. Assume the ends of the wave are
x
t
xed and that the wave is set in motion with no initial velocity from the initi
Summer 2011, Math 309 C
Quiz 2
1. Let A be a (2 2) matrix with eigenvalues r1 = a + bi and r2 = c + di.
Fill in the following chart about the system of dierential equations x = Ax.
Assume A has two linearly independent eigenvectors.
Eigenvalue type
a > 0
Summer 2011, Math 309 C
Quiz 1
011
1. a) Find the characteristic polynomial of A = 1 0 1 and check
110
that 1 and 2 are the roots to the polynomial.
b) Find a basis for the eigenspace corresponding to the eigenvalue 1.
1
Math 309C (Ward)
Final (60 Minutes)
August 19, 2011
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want g
Math 309C (Ward)
Final (60 Minutes)
August 19, 2011
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want g
Math 309C (Ward)
Midterm (60 Minutes)
July 18, 2011
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want g
Summer 2011, Math 309 C
Quiz 7
1. Suppose h : R2 R is a harmonic function. In polar coordinates you
found that h(1, ) = sin .
a) Is it possible that h is bounded (give a reason)?
No. Every bounded harmonic function is constant.
b) What is h(0, 0)?
The Mea
Math 309C (Ward)
Midterm (60 Minutes)
July 18, 2011
DO NOT TURN THIS PAGE UNTIL INSTRUCTED TO BEGIN!
Instructions.
(1) Write your name on the line provided below these instructions.
(2) Your solutions must be NEAT and LEGIBLE.
(3) Show all work you want g