21-356 Lecture 23

We say that g is conservative vector eld if there

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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.

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