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Unformatted text preview: en 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)) i (t) dt =
p! (t) dt = p (b)'p (a) = f (x)'f (y) ,
!
a i=1 # xi
a
59 where we have used the fundamental theorem of calculus.
The previous theorem shows that if a conservative vector ﬁeld is continuous,
then its integral along a curve joining two points depends only on the value at
the two points and not on the particular curve. If U is connected, then this
condition turns out to be equivalent to the vector ﬁeld being conservative.
Theorem 100 Let U ) RN be an open connected set and le...
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This document was uploaded on 03/31/2014.
 Spring '14

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