show that $\int \int_{S}^{}{curl\vec{F}\cdot \vec{dS}}$ is proportional to the lenght of $C$

The curve $ C$ is the edge of a surface S , with $ \vec{T}$ unit tangent vector of the curve $ C$ and $ \vec{F}$ a vector field such as $ \vec{F}=k\vec{T} $ for each point of $ C$ , where $ k$ is a constant , how can I prove that

$ \int \int_{S}^{}{curl\vec{F}\cdot \vec{dS}}\; $ is proportional to the length of $ C$

Any help would be appreciate . Thanks in advance.