Engineering Calculus Notes 204

Engineering Calculus Notes 204 - | t i − t i − 1 | <...

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192 CHAPTER 2. CURVES use composition to show that this assumption involves no loss of generality. Then mimic the proof in Lemma 2.4.2 that the recalibration function must be C 1 with non-vanishing derivative. ) 11. Partition into arcs: Suppose −→ p ( t ), t I is a regular vector-valued function deFned on the interval I . In this exercise, we show that I can be partitioned into closed intervals on each of which −→ p is one-to-one. ±irst we show that if I = [ a,b ] is a closed interval then there exist Fnitely many points a = t 0 < t 1 < ··· < t n = b such that the restriction of −→ p to each of the intervals [ t i 1 ,t i ] is one-to-one: (a) Assume that −→ p isn’t already one-to-one. Then for each t I there exists ε ( t ) > 0 such that −→ p is one-to-one on the interval ( t ε ( t ) ,t + ε ( t )). Show that if we pick ε ( t ) to be the largest such value, then there exist t n = t ′′ I with | t t | ≤ ε ( t ) and | t t ′′ | ≤ ε ( t ) ( with at least one equality) such that −→ p ( t ) = −→ p ( t ′′ )). (b) Now consider ε = inf t I ε ( t ). If ε > 0 then Show that any partition for which
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Unformatted text preview: | t i − t i − 1 | < ε for each i will work. (c) Prove by contradiction that ε > 0: if not, there exist pairs t ′ i n = t ′′ i with | t ′ i − t ′′ i | < 1 i −→ p ( t ′ i ) = −→ p ( t ′′ i ). By going to a subsequence, assume t ′ i → t . Show that also t ′′ i → t . But then −→ p is not locally one-to-one at t . (d) Now, let I be any interval (open, closed, or half-open). Show that there is a bisequence of points t i , i ∈ Z such that t i − 1 < t i and −→ p is one-to-one on [ t i − 1 ,t i ] ( Hint: By the preceding, we can partition any closed bounded subinterval of I into Fnitely many arcs. Use this to show that all of I can be partitioned into a possibly inFnite collection of abutting arcs.) 12. Show that the ±olium of Descartes (±igure 2.25 ) is given by the polar equation r = 3sin θ cos θ sin 3 θ + cos 3 θ ....
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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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