Smooth Curves

     Recall that the one variable function is not differentiable at because it has a "corner" at In three dimensions we have the concept of a vector function representing a smooth curve; that is, where there are no so called "corners". With an extra degree of freedom we require the derivative to not only exist but also to be continuous and nonzero.   

Definition (Smooth Curve) The graph of the vector function defined by is smooth on any interval of where is continuous and .

Example (Smooth Curve) Determine where the graph of the vector function



is smooth.

    Solution. The graph of the vector function is smooth over any interval not containing because



and for any except

Definition (Piecewise Smooth Curve) The graph of the vector function defined by is piecewise smooth on any interval that can be subdivided into a finite number of subintervals on which is smooth.

Example (Piecewise Smooth Curve) Determine where the graph of the vector function  



is piecewise smooth.

    Solution. The graph of the vector function is piecewise smooth everywhere because



and except for

Example (Piecewise Smooth Curve) Determine if the graph of the vector function  



is piecewise smooth on .

    Solution. The graph of the vector function is not piecewise smooth over because



and for where is an integer.

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Smooth Curves
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/smooth-curves.html