Let V ‰  θ be open in R2. A function F: V †’ R2 is said to be conservative on V if and only if there is a function f : V †’ R such that F = f on V. Let (x, y) ˆˆ V and let F = (P, Q) : V †’ R2 be continuous on V.
a) Suppose that C(x) is a horizontal line segment terminating at (x, y); that is, a line segment of the form L((x1, y); (x, y)), oriented from (x1, y) to (x, y). If C(x) is a subset of V, prove that

Make and prove a similar statement for Ï‘/Ï‘y and vertical line segments in V terminating at (x, y).
b) Let (x0, y0) ˆˆ V. Prove that

for all closed piecewise smooth curves C Š‚ V if and only if for all (x, y) ˆˆ V, the integrals

give the same value for all piecewise smooth curves C(x, y) which start at (x0, y0), end at (x, y), and stay inside V.
c) Prove that F is conservative on V if and only if (*) holds for all closed piecewise smooth curves C which are subsets of V.
d) Prove that if F is C1 and satisfies (*) for all closed piecewise smooth curves C which are subsets of V, then

If V is nice enough, the converse of this statement also holds (see Exercise 13.6.8).

F-T ds = P(x, y). r) F-Tds0 aJca.y) F.Tds f (x , y):= ay ax