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21-356 Lecture 23 - Monday Next we introduce the notion of...

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Next we introduce the notion of an oriented curve. Definition 95 Given a curve # with parametric representations ! : I * R N and " : J * R N , we say that ! and " have the same orientation if the para- meter 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 of parametric rep- resentations with the same orientation. Note that any curve # gives rise to two oriented curves. Indeed, it is enough to fix a parametric representation ! : I * R N and considering the equivalence class # + of parametric representations with the same orientation of ! and the equivalence class # # of parametric representations with the opposite orientation of ! . Let E ) R N and let g : E * R N be a continuous function. Given a piecewise C 1 oriented curve # with parametric representation ! : [ a, b ] * R N such that ! ([ a, b ]) ) E , we define A !
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