Suppose that E is a two-dimensional region such that if (x, y) ∈ E, then the line segments from (0, 0) to (x, 0) and from (x, 0) to (x, y) are both subsets of E. If F: E → R2 is C1 prove that the following three statements are equivalent.
a) F = f on E for some f: E → R.
b) F = (P, Q) is exact (i.e., Qx = Py on E).
c) ∫C F ∙ T ds = 0 for all piecewise smooth curves C = ϑΩ oriented counterclockwise, where Q is a two-dimensional region which satisfies the hypotheses of Green's Theorem, and Ω ⊂ E.