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21-356 Lecture 22

# G ds a a iii f ds f ds l max f

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Unformatted text preview: all a, b ! R, A (af + bg ) ds = a ! (ii) if f % g , then A ! A f ds + b ! f ds % A A g ds, ! g ds, ! .A .A . . (iii) . f ds. % |f | ds % L (+ ) max! |f |, where " is the range of # , . . ! ! (iv) if ! : [a, b] * RN is a parametric representation of # , c ! (a, b), and # 1 and # 2 are the curves of parametric representations !1 : [a, c] * RN and !2 : [c, b] * RN , then A A A f ds = f ds + f ds. ! !1 !2 Friday, March 04, 2011 Midsemester break. No classes Spring break. No classes. Monday, March 14, 2011 Next we introduce the notion of an oriented curve. Deﬁnition 95 Given a curve # with parametric representations ! : I * RN and " : J * RN , we say that ! and " have the same orientation if the parameter change h : I * J is increasing and opposite orientation if the parameter change h : I * J is decreasing. If ! and " have the same orientation, we write * ! 4 ". * Exercise 96 Prove that 4 is an equivalence relation. Deﬁnition 97 An oriented curve # is an equivalence class...
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