Engineering Calculus Notes 197

Engineering Calculus Notes 197 - 2.4. REGULAR CURVES 185...

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Unformatted text preview: 2.4. REGULAR CURVES 185 component functions has nonzero derivative on an interval of the form | t t | < for > sufficiently small; if the first component has this property, then the projection of the subcurve defined by this inequality onto the xy-plane (resp. xz-plane) is the graph of y (resp. of z ) as a C 1 function of x . From this we can conclude that, as in the planar case, the tangent line to the parametrization at any particular parameter value t = t is well-defined, and is the line in space going through the point p ( t ) with direction vector v ( t ); furthermore, the linearization of p ( t ) at t = t is a regular vector-valued function which parametrizes this line, and has first-order contact with p ( t ) at t = t . As a quick example, we consider the vector-valued function p ( t ) = (cos t, sin t, sin3 t ) with velocity v ( t ) = ( sin t, cos t, 3cos 3 t ) ....
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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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