Suppose that (mathbf{F}) is defined on (mathbf{R}^{3}) and that (oint_{C} mathbf{F} cdot d mathbf{r}=0) for all closed

Question:

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.

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