LSweek32

LSweek32 - MATH1211/LSweek32/TNK/10-11(2) THE UNIVERSITY OF...

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1 MATH1211/LSweek32/TNK/10-11(2) THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211: Multivariable Calculus Lecture Summary – Week 32 April 4, 2011 We noted that Theorem 3.2 of Section 6.3 (p.391) says that: F has path-independent line integrals is the same as H C F ¢ d s =0 for all C 1 , simple, closed curves C in the domain of F , and the requirement for this to hold is that F is a continuous vector field. Theorem 3.3 (p.392 393) says that: F has path-independent line integrals is the same as F = r f (i.e., F is a gradient field ( conservative vector field )) for some f of class C 1 , and the requirement for this to hold is that F is a continuous vector field defined on its domain of definition which is a connected , open region of R n . Moreover, when F = r f , then for any piecewise C 1 curve C in the domain of F with initial point A and terminal point B , we have R C F ¢ d s = f ( B ) ¡ f ( A ) . [A side note: in fact if F = r f for some scalar function f of class C 1 then it follows that F has path-independent line integrals, irrespective of whether the domain of F is connected or not (see bottom of p.392, before Theorem 3.3). It is the converse of this that requires that the domain of F is open and connected.] We remarked that the property of having path-independent line integrals is very nice, but it is impractical to check this directly for every possible path from every possible initial point to every possible terminal point. In this respect, Theorem 3.2 does not offer much help also, because there are infinitely many simple closed curves, and hence it is impossible to check whether H C F ¢ d s =0 for all simple closed C 1 curves C (yet, if we found that
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LSweek32 - MATH1211/LSweek32/TNK/10-11(2) THE UNIVERSITY OF...

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