2.4. REGULAR CURVES
191
with
dx
dt
negationslash
= 0 at
t
=
t
0
, and hence nearby (
i.e.
,
on
J
). Now consider
the two plane curves parametrized by
−→
p
y
(
t
) = (
x
(
t
)
,y
(
t
))
−→
p
z
(
t
) = (
x
(
t
)
,z
(
t
))
.
These are the projections of
C
onto, respectively the
xy
plane and the
xz
plane. Mimic the argument for Proposition
2
to show that each of
these is the graph of the second coordinate as a function of the first.
Challenge problems:
10.
Arcs:
In this exercise, we study some properties of arcs, as defined
in Definition
2.4.6
.
(a) Suppose
−→
p
:
I
→
R
3
is a continuous, onetoone vectorvalued
function on the closed interval
I
= [
a,b
] with image the arc
C
,
and suppose
−→
p
(
t
i
)
→
−→
p
(
t
0
)
∈C
.
Show
that
t
i
→
t
0
in
I
. (
Hint:
Show that the sequence
t
i
must have at least one accumulation
point
t
∗
in
I
, and that for
every
such accumulation point
t
∗
, we
must have
−→
p
(
t
∗
) =
−→
p
(
t
0
). Then use the fact that
−→
p
is
onetoone to conclude that the
only
accumulation point of
t
i
is
t
∗
=
t
0
. But a bounded sequence with exactly one accumulation
point must converge to that point.)
(b) Show that this property fails for the parametrization of the
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 Fall '08
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 Calculus, Topology, Metric space, accumulation point, regular curve, accumulation point t∗

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