Engineering Calculus Notes 212

Engineering Calculus Notes 212 - 200 CHAPTER 2. CURVES →...

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Unformatted text preview: 200 CHAPTER 2. CURVES → Picking − j , j = 1, . . . , n as above, we get v b a → → − (t) dt − ℓ (P , − ) = ˙ p p n Ij j =1 n ≤ Ij j =1 → − (t) dt ˙ p → − − j δj v → − ( t) − − → ˙ p vj dt n δj < 3δ j =1 = 3δ(b − a). Proof of Theorem 2.5.1. We will use the fact that since the speed is continuous on the closed interval [a, b], it is uniformly continuous, which means that given any δ > 0, we can find µ > 0 so that it varies by at most δ over any subinterval of [a, b] of length µ or less. Put differently, this says that the hypotheses of Lemma 2.5.2 are satisfied by any partition of mesh size µ or less. We will also use the easy observation that refining the → partition raises (or at least does not lower) the “length estimate” ℓ (P , − ) p associated to the partition. Suppose now that Pk is a sequence of partitions of [a, b] for which → ℓk = ℓ (Pk , − ) is strictly increasing with limit s (C ) (which, a priori may p be infinite). Without loss of generality, we can assume (refining each partition if necessary) that the mesh size of Pk goes to zero monotonically. Given ε > 0, we set ε δ= 3(b − a) and find µ > 0 such that every partition with mesh size µ or less satisfies the hypotheses of Lemma 2.5.2; eventually, Pk satisfies mesh(Pk ) < µ, so b a − (t) dt − ℓ (P , − ) < 3δ(b − a) = ε. → → ˙ p kp This shows first that the numbers ℓk converge to assumption, lim ℓk = s (C ), so we are done. b a → − (t) dt—but by ˙ p The content of Theorem 2.5.1 is encoded in a notational device: given a → regular parametrization − : R → R3 of the curve C , we define the p ...
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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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