Let Icirc copy be a three dimensional region and F Q R3 be C1 on Q Suppose further that, for each (x, y, z) Icirc copy , both the line segments L((x, y, 0) (x, y, z)) and L((x, 0, 0) (x, y, 0)) are subsets of Icirc copy Prove that the following statements are equivalent a) There is a C2 function G ...

The proof that i implies ii and ii implies iii is similar to the proof of Theorem 136 ...

Let Î© be a three-dimensional region and F: Q R3 be C1 on Q. Suppose further that,

Question:

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