In a rotating frame the (unforced, incompressible) Euler equation is $$ \frac{∂\vec{u}}{∂t}+\vec{u}\cdot\nabla\vec{u}=-\nabla\left(\frac{p}{\rho_0}\right)-2\vec{\Omega}\times\vec{u}+\nabla\left(\frac{1}{2}|\vec{\Omega}\times\vec{x}|^2\right) $$ where $\vec{\Omega}$ is a constant vector (the angular velocity).
Taking the curl of this equation (and using suitable vector identities), I can show that the vorticity $\vec{\omega}=\nabla\times\vec{u}$ satisfies $$ \frac{∂\vec{\omega}}{∂t}=\nabla\times\left(\vec{u}\times\vec{\omega}\right)+2\nabla\times\left(\vec{u}\times\vec{\Omega}\right) $$ Now I want to show that, if $S_t$ is a material surface within the fluid, then the following is true $$ \frac{d}{dt}\int_{S_t}{\left(\vec{\omega}+2\vec{\Omega}\right)\cdot d\vec{S}}=0 $$ I can write $$ \vec{\omega}+2\vec{\Omega}=\nabla\times\vec{u}+\nabla\times\left(\vec{\Omega}\times\vec{x}\right) $$ and use Stokes' theorem to rewrite this as a line integral $$ \frac{d}{dt}\int_{S_t}{\left(\vec{\omega}+2\vec{\Omega}\right)\cdot d\vec{S}}=\frac{d}{dt}\int_{∂S_t}\left(\vec{u}+\vec{\Omega}\times\vec{x}\right)\cdot d\vec{l} $$ but here is where I am stuck. I am given the hint (for any material curve $\gamma_t$) $$ \frac{d}{dt}\int_{\gamma_t}\vec{a}\cdot d\vec{l}=\int_{\gamma_t}\frac{D\vec{a}}{Dt}\cdot d\vec{l}+\int_{\gamma_t}a_i\left(∂_ju_i\right)dl_j $$ But when I substitute $\vec{a}=\vec{u}+\vec{\Omega}\times\vec{x}$ here nothing seems to become clearer.