9. Let 0. be a three-dimensional region and F : 0. -4 R3 be C1 on n. Suppose further that for each (x,y,z) En, both the line segments L((x,y,O); (x,y,z))

and L((x, 0, 0); (x, y, 0)) are subsets of n. Prove that the following statements are equivalent.

(a) There is a C2 function G: 0. -4 R3 such that curlG = F on n.

(b) If F, E, and S = aE satisfy the hypotheses of Gauss's Theorem and E C 0., then lis F ·nder = 0,

(c) The identity div F = ° holds everywhere on n.