Engineering Calculus Notes 199

Engineering Calculus Notes 199 - 187 2.4. REGULAR CURVES...

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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 vector-valued function − (t) is piecewise regular on p the interval I if → 1. − (t) is continuous on I , p → 2. − (t) is locally one-to-one 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 piecewise-regular vector-valued 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 well-defined derivative from the left and and a (possibly different) derivative from the right. Points where the two one-sided derivatives are different are called corners (for example the vertices of a polygon) or, in the extreme case where the two one-sided derivatives point in exactly opposite directions (such as the points where a cycloid hits the x-axis), cusps. Exercises for § 2.4 Practice problems: → → 1. For each pair of vector-valued 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.

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