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Unformatted text preview: 191 2.4. REGULAR CURVES with dx = 0 at t = t0 , and hence nearby (i.e., on J ). Now consider
the two plane curves parametrized by
− (t) = (x(t) , y (t))
− (t) = (x(t) , z (t)).
z 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 − : I → R3 is a continuous, one-to-one vector-valued
function on the closed interval I = [a, b] with image the arc C ,
and suppose − (ti ) → − (t0 ) ∈ C . Show that ti → t0 in I . (Hint:
Show that the sequence ti must have at least one accumulation
point t∗ in I , and that for every such accumulation point t∗ , we
must have − (t∗ ) = − (t0 ). Then use the fact that − is
one-to-one to conclude that the only accumulation point of ti is
t∗ = t0 . 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
circle by − (θ ) = (cos θ, sin θ ), 0 ≤ θ < 2π .
p (c) Show that the graph of a continuous function f (x) deﬁned on a
closed interval [a, b] is an arc. (d) Give an example of an arc in space whose projections onto the
three coordinate planes are not arcs. (This is a caution
concerning how you answer the next question.)
(e) In particular, it follows from Proposition 2.4.7 that every
regular curve in the plane is locally an arc. Show that every
regular curve in space is also locally an arc.
(f) Show that, as in Lemma 2.4.2, any two regular parametrizations
of the same arc are reparametrizations of each other. (Hint:
First, prove that if one of the parametrizations is assumed to be
globally one-to-one, and deﬁned on [a, b], then there is a unique
recalibration function relating the two parametrizations; then ...
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.
- Fall '08