Engineering Calculus Notes 199

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

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2.4. REGULAR CURVES 187 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 −→ p ( t ) is piecewise regular on the interval I if 1. −→ p ( t ) is continuous on I , 2. −→ p ( t ) is locally one-to-one on I , 3. there is a partition P such that the restriction of −→ p ( t ) to each closed atom [ p i ,p i +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
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