21-356 Lecture 23

We say that g is conservative vector eld if there

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

This document was uploaded on 03/31/2014.

Ask a homework question - tutors are online