x
0
, y
0
) = 0 and the discriminant
d
=
f
xx
(
x
0
, y
0
)
f
yy
(
x
0
, y
0
)

f
xy
(
x
0
, y
0
)
2
is nonzero (i.e., the Second Derivative Test does not
fail).
a) Prove that if
u
(
x, y
) satisFes 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
) satisFes 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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View Full Documentc) Provide an example of a function
u
(
x, t
) that satisFes 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 ±ourier series for
f
(
x
) =

x

on

π
≤
x < π
is:
π
2

4
π
∞
s
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 inFnite sum identity.
b) ±ollowing the process you performed in 2.57 b) and c), use the value of
ζ
(4) to come up
with the same inFnite sum identity.
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 Spring '11
 NormanKatz
 Critical Point, Partial differential equation, inﬁnite sum identity

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