\[A,B\]
on a closed curve \[C\]
.
\[\int_C \mathbf{F} \cdot d \mathbf{r}= \int_{\gamma_1} \mathbf{F} d \mathbf{r} + \int_{\gamma_2} \mathbf{F} d \mathbf{r}\]
\[\gamma_1 \]
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.