cbicon.png CALCULUS Understanding Its Concepts and Methods

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Curve, Smooth, Piecewise smooth

A curve is a continuous vector-valued function r(t)  whose domain is an interval [a, b]. A plane curve has values in $\QTR{Bbb}{R}^{2}$; a space curve has values in $\QTR{Bbb}{R}^{3}$.

A curve r(t) is

smooth if r'(t)  is continuous and never equal to the zero vector.

piecewise smooth if it consists of a finite number of smooth pieces.

simple if MATH whenever MATH.

closed if r(a) = r(b .

graphics/IE_curve__11.png graphics/IE_curve__12.png graphics/IE_curve__13.png
Smooth Piecewise smooth Simple, not closed
graphics/IE_curve__14.png graphics/IE_curve__15.png graphics/IE_curve__16.png
Closed, not simple Not closed, not simple Simple, closed


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Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.

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