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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.
 Spring '14

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