Suppose that \(\mathbf{F}\) is defined on \(\mathbf{R}^{3}\) and that \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}=0\) for all closed paths \(C\) in \(\mathbf{R}^{3}\). Prove:
(a) \(\mathbf{F}\) is path independent; that is, for any two paths \(C_{1}\) and \(C_{2}\) in \(\mathcal{D}\) with the same initial and terminal points,
\[
\int_{C_{1}} \mathbf{F} \cdot d \mathbf{r}=\int_{C_{2}} \mathbf{F} \cdot d \mathbf{r}
\]
(b) \(\mathbf{F}\) is conservative.