Let \(\mathbf{F}\) be a vector field on an open, connected domain \(\mathcal{D}\) with continuous second partial derivatives. Which of the following statements are always true, and which are true under additional hypotheses on \(\mathcal{D}\) ?
(a) If \(\mathbf{F}\) has a potential function, then \(\mathbf{F}\) is conservative.
(b) If \(\mathbf{F}\) is conservative, then the cross-partial derivatives of \(\mathbf{F}\) are equal.
(c) If the cross-partial derivatives of \(\mathbf{F}\) are equal, then \(\mathbf{F}\) is conservative.