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Unformatted text preview: 187 2.4. REGULAR CURVES Piecewise Regular Curves
Deﬁnition 2.4.1 applies to most of the curves we consider, but it excludes a
few, notably polygons like triangles or rectangles and the cycloid
(Figure 2.21). These have a few exceptional points at which the tangent
vector is not well deﬁned. For completeness, we formulate the following
→
Deﬁnition 2.4.9. A vectorvalued function − (t) is piecewise regular on
p
the interval I if
→
1. − (t) is continuous on I ,
p →
2. − (t) is locally onetoone on I ,
p
→
3. there is a partition P such that the restriction of − (t) to each closed
p
atom [pi , pi+1 ] is regular.
A curve C is piecewise regular if there is a piecewiseregular
vectorvalued function which parametrizes C .
The requirement that the restriction to each closed atom is regular
requires, in particular, that at every partition point there is a welldeﬁned
derivative from the left and and a (possibly diﬀerent) derivative from the
right. Points where the two onesided derivatives are diﬀerent are called
corners (for example the vertices of a polygon) or, in the extreme case
where the two onesided derivatives point in exactly opposite directions
(such as the points where a cycloid hits the xaxis), cusps. Exercises for § 2.4
Practice problems:
→
→
1. For each pair of vectorvalued functions − (t) and − (t) below, ﬁnd a
p
q
→
− (s)=− (t(s)) and another, s(t),
→
recalibration function t(s) so that q
p
→
→
so that − (t)=− (s(t)):
p
q
(a)
→
− (t) = (t, t) − 1 ≤ t ≤ 1
p
→
− (t) = (cos t, cos t) 0 ≤ t ≤ π
q
(b)
→
− (t) = (t, et ) − ∞ < t < ∞
p
→
− (t) = (ln t, t) 0 < t < ∞
q ...
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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
 ALL
 Calculus, Angles, Polygons

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