7. Let V =f. 0 be open in R2. A function F : V --+ R 2 is said to be conservative on V if and only if there is a function f : V --+ R such that F = '\l f on V. Let

(x, y) E 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), i.e., a line segment of the form L((X1, y); (x, y)), oriented from (Xl, y) to (x, y). If C(x) is a subset of V, prove that

~ 1 F· Tds = P(x,y).

uX c(x)

Make and prove a similar statement for 0/ oy and vertical line segments in V terminating at (x, y).

(b) Let (xo, Yo) E V. Prove that

(*) fa F·Tds=O