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Unformatted text preview: 200 CHAPTER 2. CURVES →
Picking − j , j = 1, . . . , n as above, we get
− (t) dt − ℓ (P , − ) =
Ij j =1
n ≤ Ij j =1 →
− (t) dt
− − j δj
− ( t) − −
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 , − )
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
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
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
a − (t) dt − ℓ (P , − ) < 3δ(b − a) = ε.
kp This shows ﬁrst that the numbers ℓk converge to
assumption, lim ℓk = s (C ), so we are done. b
− (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
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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