CALCULUS Understanding Its Concepts and Methods

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
;
a space curve has values in
.

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 whenever .

closed
if
**r**(*a*) = **r**(*b* .

Smooth | Piecewise smooth | Simple, not closed | |

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.