a) Prove that if
u
(
x, y
) satisfies the Laplace equation (no boundary conditions are assumed),
then
u
(
x, y
) has no nondegenerate minima or maxima, i.e., all nondegenerate critical points
are saddle points.
b) Prove that if
u
(
x, t
) satisfies the heat equation (no boundary conditions are assumed),
then
u
(
x, t
) has no nondegenerate minima or maxima, i.e., all nondegenerate critical points
are saddle points.
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c) Provide an example of a function
u
(
x, t
) that satisfies the wave equation AND it has a
relative minimum and/or a relative maximum. As with the previous problems, no boundary
conditions are assumed.
7)
(10 points) You showed in Homework 2 that the Fourier series for
f
(
x
) =

x

on

π
≤
x < π
is:
π
2

4
π
∞
summationdisplay
k
=0
1
(2
k
+ 1)
2
cos(2
k
+ 1)
x
a) Use Parseval’s Identity on
f
(
x
) =

x

to come up with a neat infinite sum identity.
b) Following the process you performed in 2.57 b) and c), use the value of
ζ
(4) to come up
with the same infinite sum identity.
c) Using the Fourier series itself, prove that
∞
summationdisplay
k
=0
1
(2
k
+ 1)
2
=
π
2
8
8)
(10 points)
a) Let
a
=
b
= 1,
f
1
(
x
) = 0,
f
2
(
x
) =
x
,
g
1
(
y
) = 0, and
g
2
(
y
) =
y
.
Solve the Laplace
equation with boundary conditions
u
(
x,
0) =
f
1
(
x
),
u
(
x,
1) =
f
2
(
x
),
u
(0
, y
) =
g
1
(
y
), and
u
(1
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 Spring '11
 NormanKatz
 Critical Point, Partial differential equation, inﬁnite sum identity

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