8. Let E be a two-dimensional region such that if (x, y) E E, then the line segments from (0,0) to (x,O) and from (x,O) to (x, y) are both subsets of E. If F: E -4 R2 is C1, prove that the following three statements are equivalent.

(

a) F = \l! on E for some ! : E -4 R.

(b) F = (P,Q) is exact; i.e., Qx = Py on E.

(c) J c F . T ds = ° for all piecewise smooth curves C = an oriented counterclockwise, where 0. is a two-dimensional region that satisfies the hypotheses of Green's Theorem, and 0. c E.