MATH 234 NWU MULTIPLE INTEGRATION AND VECTOR CALCULUS PRACTICE QUIZ Q & A

1. (10+5 points) Let F = P i + Qj be a vector field on a domain D in the plane. (a) Give two different characterizations for the vector field F to be conservative which are valid on ANY domain D. Solution. There are three different possible characterizations: – F = ∇f for some scalar function f. – F ds = 0 for all closed curves C in the domain of F. – For any curve C between points p and q, C F · ds depends only on the endpoints (p and q), not on the curve itself. (Path independence). (b) Give a condition for F to be conservative on a simply connected domain D in terms of the derivatives of P and Q. Solution. ∇ × F = 0. Let C be a closed curve in the domain D, and let S be the region in D bounded by C. Then,since D (the domain of F) issimply connected, F will be defined on all of S. So, by Green’s Theorem, ∫∫ S ∇ × F · kdA = I F · ds. If ∇ × F = 0, then for any closed curve C in the domain, C F · ds = 0, so F is conservative. (c) Give an example to show that the condition in (b) is not sufficient if D is not simply connected. Solution. Let F = −yi + xj , x 2 + y 2 with the domain of F being R2 − {(0, 0)}, i.e., the plane minus the origin (F is not defined at the origin). Then, ∇ × F = 0, but F is not conservative