Proof That Integral of the Curl of a Vector Field Tangential to a Surface is Zero About the Border of The Surface

Theorem
Let  
\[\mathbf{F}\]
  be a vector field with differentiable components defined on a surface  
\[S\]
, such that  
\[\mathbf{\nabla} \times \mathbf{F}\]
  is tangential to  
\[S\]
  everywhere on  
\[S\]
.
If  
\[C\]
  is the border of  
\[S\]
  then  
\[\oint_C \mathbf{F} \cdot d \mathbf{r} =0\]

Proof
Stoke's Theorem states  
\[\oint_C \mathbf{F} \cdot d \mathbf{r} = \int \int_S (\mathbf{\nabla} \times \mathbf{F}) \cdot \mathbf{n} dS \]

Since  
\[\mathbf{\nabla} \times \mathbf{F}\]
  is tangential to  
\[S\]
  everywhere,  
\[\mathbf{\nabla} \times \mathbf{F}=0\]
  everywhere on  
\[S\]
.
Hence  
\[\oint_C \mathbf{F} \cdot d \mathbf{r} =0\]
.