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.

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