## Proof That Integral Between Two Points is Independent of Path If the Integral of a Vector Field Around Any Closed Path is Zero

Take any two points
$A,B$
on a closed curve
$C$
.

Then
$\int_C \mathbf{F} \cdot d \mathbf{r}= \int_{\gamma_1} \mathbf{F} d \mathbf{r} + \int_{\gamma_2} \mathbf{F} d \mathbf{r}$
where
$\gamma_1$
is the part of the curve
$C$
from
$A$
to
$B$
shown above and
$\gamma_2$
is the part of the curve
$C$
from
$B$
to
$A$
nabove.
Since the integral of
$\mathbf{F}$
around any closed curve is zero,
$0=\int_C \mathbf{F} \cdot d \mathbf{r}= \int_{\gamma_1} \mathbf{F} \cdot d \mathbf{r} + \int_{\gamma_2} \mathbf{F} \cdot d \mathbf{r} \rightarrow \int_{\gamma_1} \mathbf{F} \cdot d \mathbf{r} =- \int_{\gamma_2} \mathbf{F} \cdot d \mathbf{r} \rightarrow \int_{\gamma_1} \mathbf{F} \cdot d \mathbf{r} = \int_{- \gamma_2} \mathbf{F} \cdot d \mathbf{r}$

Notice that
$\gamma_1 , - \gamma_2$
are any paths from A to B, so the the integral is independent of the paths.