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Engineering Calculus Notes 197

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

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2.4. REGULAR CURVES 185 component functions has nonzero derivative on an interval of the form | t t 0 | for ε> 0 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 0 is well-defined, and is the line in space going through the point −→ p ( t 0 ) with direction vector −→ v ( t 0 ); furthermore, the linearization of −→ p ( t ) at t = t 0 is a regular vector-valued function which parametrizes this line, and has first-order contact with −→ p ( t ) at t = t 0 . As a quick example, we consider the vector-valued function −→ p ( t ) = (cos t, sin t, sin 3 t ) with velocity −→ v ( t ) = ( sin t, cos t, 3 cos 3 t ) . Since sin t and cos t cannot both be zero at the same time, this is a regular parametrization of a curve in space, sketched in Figure 2.26 . We note, for
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