lecture notes 13

.Proposition. Let F be a continuous vector field defined on a region D,prove ∫ that the line integral of F is independent of path if and only ifF · T ds = 0 for any piecewise smooth closed curve C in D.. CProof. Suppose that line integral of F is independent of path, then letC be any closed curve with the same starting and terminal point A,then the constant path C ′ with A for all t is also a curve with the samestarting and terminal point ∫ A. It follows from the path ∫ independence ofthe line integral of F that F · T ds =C∫C F · T ds = F · 0 ds = 0.′ C ′Conversely, suppose C 1 and C 2 are two paths, both of them startsfrom the same point A, and terminates at point B. Let C = C 1 ∪ (−C 2 )be a closed path from A to B via C 1 , and back from B to A via −C 2 (inreverse direction of C 2 ). Then C is a piecewise smooth closed curvein D, ∫hence one has ∫∫∫∫0 = F · T ds = F · T ds + F · T ds = F · T ds − F · T ds.C ∫C 1 ∫−C 2 C 1 C 2Hence F · T ds = F · T ds. So the line integral of F isC 1 C 2independence of path.. . . . . .