MAT 3320 Example of Mid Term Exam
October 4, 2004
Duration: 80 minutes
INSTRUCTIONS: This is a closed book exam. No calculator of any sort is
allowed. Justify all your responses. You do not need to prove theorems from your notes
unless asked to do so, but
MAT 3320 Assignment 3: Solutions
Note: You HAD to justify all computations.
1. (a) After you get rid of the window dressing, the problem boils down to solving the partial
dierential equation
2
2m
2
2
+ 2
x2 y
(x, y) = E(x, y),
together with the boundary c
MAT 3320 Assignment 2: Solutions
Note: You HAD to justify all computations.
1. As seen in class, you either get
f (x) k 2 f (x) = 0
and g (t) k 2 c2 g(t) = 0,
or
f (x) = 0
and g (t) = 0.
In the rst case, we have (using ODE technics)
fk (x) = Aekx + Bekx
a
MAT 3320 Assignment 4: Solutions
Note: You HAD to justify all computations.
1. Since xo is an ordinary point, P (x) and Q(x) are analytic at xo . That means that they
can be expanded in a power series about xo , i.e.,
an (x xo )n
P (x) =
and
Q(x) =
n=0
bn
MAT 3320 Assignment 2: Due November 8, 2004
Note: You have to justify all computations.
1. When solving the one-dimensional wave equation for a string of length L with both ends
xed
2y
1 2y
= 2 2
2
x
c t
using the separation of variable process, as seen i
MAT 3320 Assignment 1: Solutions
Note: You HAD to justify all computations.
1. The only tools allowed are the ones dening a normed vector space. Thus assuming that
there is an inner product is wrong.
We use the good old add zero trick:
f
=
f +0
=
(f g) +
MAT 3320 Mid Term Exam
October 21, 2004
Duration: 80 minutes
INSTRUCTIONS: This is a closed book exam. No calculator of any sort is
allowed. Justify all your responses. You do not need to prove theorems from your notes
unless asked to do so, but you must
MAT 3320 Mid Term Exam: Solutions
1. (a) An inner product space (V ; <, >) is a K-vector space V (K = R or C) equipped
with a mapping
<, >: V V K
such that
i. < v, v > R and 0 for all v V ; moreover < v, v >= 0 if and only if
v = 0.
ii. < v, w >= < w, v >
MAT 3320 Denitions
1. A K-vector space: A vector space V over some scalar (eld) K(= R or C), is a set
equipped with two operations
+:V V V
and : K V V
where the two operations are required to satisfy to the following eight axioms, i.e., for all
A, B, C V
MAT 3320 Assignment 1: Due October 14, 2004
Note: You have to justify all computations.
1. Given a normed vector space (V, ), show that
| f g | f g
for all f, g V .
2. Let C[a, b] be the set of complex-valued functions on [a, b]. Verify that it is a vecto
MAT 3320 Assignment 3: Due November 25, 2004
Note: You have to justify all computations.
1. A particle is forced to live within a rectangular plate (2-dimensional), i.e., it is under the
inuence of the following potential:
V (x, y) =
0 0 x a and 0 y b
oth