Let Ω be a three-dimensional region and F: Q R3 be C1 on Q. Suppose further that, for each (x, y, z) Ω, both the

Let Ω be a three-dimensional region and F: Q †’ R3 be C1 on Q. Suppose further that, for each (x, y, z) ˆˆ Ω, both the line segments L((x, y, 0); (x, y, z)) and L((x, 0, 0); (x, y, 0)) are subsets of Ω. Prove that the following statements are equivalent.
a) There is a C2 function G: Ω †’ R3 such that curl G = F on Ω.
b) If F, E, and S = Ï‘E satisfy the hypotheses of the Divergence Theorem and E Š‚ Ω, then

c) The identity div F = 0 holds everywhere on Ω.

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