APPM 2350
HOMEWORK 04
FALL 2010
Due by 4 PM on Friday, September, 24, 2010 in your TA’s mailbox
1. Work all of the “suggested” textbook problems. Remember that you do not need to write them up or
submit them. Just be sure you can do, and understand, all of them prior to starting the homework
problems that you submit for grading.
2. If you have not ±nished problem 1, go back and ±nish it.
3. Suppose that a smooth curve
r
(
t
) has the parameterization
r
(
t
)=(
x
(
t
)
,y
(
t
))
for
t
∈
R
.
We will assume that this parameterization is in terms of arclength. For a given point
t
0
∈
R
,w
e
de±ne the function
K
(
t
) as
K
(
t
)=
N
(
t
0
)
·
(
r
(
t
0
)
−
r
(
t
))

r
(
t
0
)
−
r
(
t
)

2
.
Clearly,
K
(
t
) is continuous when
t
°
=
t
0
. Our goal is to investigate the continuity of
K
(
t
)when
t
=
t
0
.
a. Show that for the parameterization given above, we can write the curvature
κ
(
t
) as
κ
(
t
)=

x
°
(
t
)
y
°°
(
t
)
−
y
°
(
t
)
x
°°
(
t
)

(
x
°
(
t
)
2
+
y
°
(
t
)
2
)
3
/
2
.
b. Letting
θ
equal the angle between
N
(
t
0
) and
r
(
t
)
−
r
(
t
0
), show that we can simplify
K
(
t
)to
K
(
t
)=
cos(
θ
)

r
(
t
0
)
−
r
(
t
)

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 Fall '07
 ADAMNORRIS
 Calculus, Laplace’s Equation

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