21-356 Lecture 23

# Then the following conditions are equivalent i g is a

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Unformatted text preview: t g : U * RN be a continuous function. Then the following conditions are equivalent. (i) g is a conservative vector ﬁeld, (ii) for every x, y ! U and for every two piecewise C 1 oriented curves # 1 and # 2 with parametric representations -1 : [a, b] * RN and -2 : [c, d] * RN , respectively, such that -1 (b) = -2 (d) = x, -1 (a) = -2 (c) = y, and -1 ([a, b]) , -2 ([c, d]) ) U , A A g= g. !1 !2 (iii) for every piecewise C 1 closed oriented curve # with range contained in U , A g = 0. ! 60 Wednesday, March 16, 2011 Proof. We prove that (i) implies (ii). Assume that g is a conservative vector ﬁeld with scalar potential f : U * R, let x, y ! U and let -1 : [a, b] * RN and -2 : [c, d] * RN be as in (ii). Then by the previous theorem A A A A g= ,f = f (x) ' f (y) = ,f = g. !1 !1 !2 !2 Conversely assume that (ii) holds. We need to ﬁnd a scalar potential for g. Fix a point x0 ! U and for every x ! U deﬁne A f (x) := g, ! 1 where # a piecewise C oriented curve with parametric representation ! : [a, b] * RN such that ! (b) = x, ! (a) = x0 , and ! ([a, b]) ) U . We claim that there exist #f (x) = gi (x) . # xi Since U is open...
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## This document was uploaded on 03/31/2014.

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