Answer
(a) the statement is always True
(b) the statement is always True
(c) the statement is true under additional hypotheses that ${\bf{F}}$ be a vector field on a simply connected domain $\cal D$
Work Step by Step
(a) Let $f$ be the potential function for ${\bf{F}}$ such that ${\bf{F}} = \nabla f$. By definition, ${\bf{F}}$ is conservative. Thus, this statement is always True.
(b) By Theorem 1 in Section 17.1, every conservative vector field satisfies the cross partials condition, that is, the cross partials of ${\bf{F}}$ are equal. Therefore, this statement is always True.
(c) By Theorem 4, this statement is true under additional hypotheses that ${\bf{F}}$ be a vector field on a simply connected domain $\cal D$.
