Does Faraday-Maxwell equation exclusively refer to a variable magnetic field instead to a variable magnetic fux? Faraday's law says that a variable magnetic flux produces an induced emf. A varying flux can be due to a varying magnetic field or a varying surface. The Faraday-Maxwell equation refers to a curl electric field produced by a variable magnetic field. Does this equation exclude the emf induced by a constant magnetic field and a variable surface? That is, is this equation less general than Faraday's law?
 A: In a given reference system, the  Maxwell-Faraday differential equation
$$
\nabla \times {\bf E} = -\frac{\partial {\bf B}}{\partial t}
$$
only implies the integral  relation
$$
\int_{\partial \Sigma}{\bf E}\cdot{\mathrm d}{\bf l}=- \int_{\Sigma}\frac{\partial}{\partial t}{\bf B}\cdot{\mathrm d}{\bf S},\tag{1}
$$
where $\Sigma$ is a surface and $\partial \Sigma$ its boundary (a closed curve in the space).
However, it is possible to rewrite this formula in terms of variations of flux due to a time variation of $\Sigma$, by noticing that the variation of the flux of ${\bf B}$ at constant ${\bf B}$ can be written as a line integral over $\partial \Sigma$:
$$
\int_{\partial \Sigma}{\bf v}\times {\bf B}\cdot{\mathrm d}{\bf l}=- \frac{{\mathrm d}}{{\mathrm d} t}\int_{\Sigma(t)}{\bf B}\cdot{\mathrm d}{\bf S}.\tag{2}
$$
The velocity ${\bf v}$ is the local value of the velocity of the line element ${\mathrm d}{\bf l}$. By combining the two formulas $(1)$ and $(2)$, we get the formula
$$
\int_{\partial \Sigma}\left({\bf E}+{\bf v}\times {\bf B})\right)\cdot{\mathrm d}{\bf l}=- \frac{{\mathrm d}}{{\mathrm d} t}\int_{\Sigma(t)}{\bf B}(t)\cdot{\mathrm d}{\bf S},\tag{3}
$$ where the left hand side is the emf due to the non-conservative electric field induced by the  time variation of ${\bf B}$ and a term that could be re-interpretated as the circuitation of the Lorentz force per unit charge over $\partial \Sigma$.
Therefore, the original integral form of the Faraday-Lenz formula can be derived from the Faraday-Maxwell partial derivative equation and it should not be considered a more general equation.
