I have been recently studying Maxwell Equations, and I wasn't able to understand properly the EMF $\zeta$. Mathematically we have $\zeta=-\dfrac{d\Phi_B}{dt}$ where $\Phi_B$ is the magnetic flux of a surface $\Sigma$.
At first, we were taught that also we have: $\zeta=\oint_{\partial\Sigma} \overrightarrow{E}\cdot \overrightarrow{dl}$ where $\overrightarrow{E}$ is the electric field and $\overrightarrow{dl}$ is an element of $\partial\Sigma$, and finally that: $\zeta=\oint_{\partial\Sigma} (\overrightarrow{v}\times\overrightarrow{B})\cdot \overrightarrow{dl}$ where $\overrightarrow{B}$ is the magnetic field and $\overrightarrow{v}$ is the velocity (is it the velocity of an element of $\partial\Sigma$?).
But after some research, I think that the first formula is used when $\Sigma$ is in rest frame (when it's not moving), and the second one is used when $\Sigma$ is moving and $\overrightarrow{B}$ is constant in respect to time.
So can you please give me a formula that generalize these two formulas? and can we derive it from$\zeta=-\dfrac{d\Phi_B}{dt}$