Prove that if $$\oint_C \vec{B}.d\vec{r}=0$$ for any closed curve $C$, then the path intergral $$\int_P^Q\vec{B}.d\vec{r}$$ is independent of the path taken between points $P$ and $Q$.
Any help would be appreciated.
2026-03-29 08:16:11.1774772171
Prove for any closed curve $C$ then the path integral is independent of the path between points $p$ and $q$
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Outline: Let $\gamma_1,\gamma_2$ be two paths from $P$ to $Q$. Then the path made by following $\gamma_1$, then $\gamma_2$ traversed in the opposite direction (often written $\gamma_2^{-1}\gamma_1$ or $\gamma_1-\gamma_2$) is a closed curve. Also, by additivity in the path, $\int_{\gamma_1-\gamma_2} = \int_{\gamma_1}-\int_{\gamma_2}$.