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Unformatted text preview: p ( t ) speciFes a curve by tracing it out. This approach is particularly useful for specifying curves in space, but as we shall see it is also a natural setting (even in the plane) for applying calculus to curves. However, it has the intrinsic complication that a given curve can be traced out in many dierent ways. In this section, we study how dierent vector-valued functions specify the same curve, and which properties of a function encode geometric properties of the curve it traces out, independent of which particular function is used to describe it. Along the way, we will formulate more carefully which kinds of vector-valued functions are appropriate parametrizations of a curve, and hopefully end up with a better understanding of what, exactly, constitutes a curve....
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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.
- Fall '08