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21-356 Lecture 22

# Indeed it is enough to x a parametric representation i

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Unformatted text preview: of parametric representations with the same orientation. Note that any curve # gives rise to two oriented curves. Indeed, it is enough to ﬁx a parametric representation ! : I * RN 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 ) RN and let g : E * RN be a continuous function. Given a piecewise C 1 oriented curve # with parametric representation ! : [a, b] * RN such that ! ([a, b]) ) E , we deﬁne A g := ! A N b+ gi (! (t)) -! (t) dt. i a i=1 Deﬁnition 98 Let U ) RN be an open set and let g : U * RN . We say that g is conservative vector ﬁeld if there exists a di!erentiable function f : U * R such that ,f (x) = g (x) for all x ! U . The function f is called a scalar potential for g. Theorem 99 (Fundamental Theorem of Calculus for Curves) Let U ) RN be an open set, let f ! C 1 (U ), let x, y ! U and let # a piecewise C 1 oriented curve with parametric representation ! : [a, b] * RN such that ! (b) = x, ! (a) = y, and ! ([a, b]) ) U . Then A ,f = f (x) ' f (y) . ! Proof. Deﬁne p (t) := f (! (t)) and observe that by Theorem 50, p is piecewise C 1 with N + #f p! (t) = (! (t)) -! (t) i # xi i=1 for all but ﬁnitely many t. Hence, A A b+ Ab N #f ,f = (! (t)) -! (t) dt = p! (t) dt = p (b)'p (a) = f (x)'f (y) , i ! a i=1 # xi a...
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