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Unformatted text preview: parametrizations via vectorvalued functions, we must Frst reformulate the basic notion of convergence, as well as dierentiation and integration, in these contexts. Convergence of Sequences of Points The convergence of sequences of points { p i } in R 2 or R 3 is a natural extension of the corresponding idea for numbers, or points on the line R . We will state everything in terms of R 3 , but the corresponding statements and/or proofs for R 2 are easy modiFcations of the R 3 versions. Before formulating a geometric deFnition of convergence, we note a few properties of the distance function on R 3 . The Frst property will allow us to use estimates on coordinates to obtain estimates on distances, and viceversa . Lemma 2.3.1. Suppose P,Q R 3 have respective (rectangular) coordinates ( x 1 ,y 1 ,z 1 ) and ( x 2 ,y 2 ,z 2 ) . Let := max(  x  ,  y  ,  z  ) (where x := x 2 x 1 , etc.)...
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 Fall '08
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 Calculus

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