i'm trying to work with the material derivative $$\frac{D}{Dt}=\frac{\partial}{\partial t}+(\vec{u}\cdot \vec{\nabla}) $$ of the circulation $\Gamma=\oint_{C=\partial S}\vec{u}\cdot d\vec{l}$, that can be expressed like $\Gamma=\iint_{S}\vec{\omega}\cdot d\vec{A}$. In Kelvins theorem, the material Derivative conmutes with the line integral around a closed contour, that it seems not vary over time (the closed curve). If i assume that it conmutes with the surface integral, this will lead me to $$\frac{D\Gamma}{Dt}=\iint_{S}\frac{D\vec{\omega}}{Dt}\cdot d\vec{A}+\iint_{S}\vec{\omega}\cdot \frac{D(d\vec{A})}{Dt}.$$
My problem is that i do not know how to treat the second term. The surface is arbitrary (with the restriction that it's boundary has to be C), and the variation of the normal to the surface is a priori unknown.
How can i solve this?