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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