## 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
$\oint_C \mathbf{F} \cdot d \mathbf{r} =0$
.