Gauss Theorem and Faraday's Law I'm studying magnetic fields and I have a big problem. The Gauss theorem states that the magnetic flux through a closed surface is null since the line of flux enter and exit from it in the same amount. However, the Faraday law talks about the flux linkage with a surface which can be not null in case, for example, the magnetic field is not constant in time. Why is not null as well, since it is the flux through a surface?
 A: Faraday's law talks about the flux piercing through an open surface, while Gauss' law for magnetic fields is for a closed surface.
A: The two concerned equations
\begin{align}\mathbf \nabla \times \mathbf E  &= -\frac{\partial{\mathbf B}}{\partial t}\tag 1 \\ \mathbf \nabla \cdot \mathbf  B &= 0\tag 2\end{align}
are conveying totally different meanings.
While the former $(1)$ correlates the circulation of the induced electric field with the negative time rate of change of the associated magnetic field; the later viz. $(2)$ says that the divergence of magnetic field is essentially zero.
Let's come to the present context where OP seems to confused with the integral forms of the equation:
The integral forms respectively are:
\begin{align} \int_\textrm C \mathbf E \cdot \mathrm d\mathbf s &= -\frac{\mathrm d}{\mathrm dt}\, \underbrace{\color{red}{\int _\mathrm S\mathbf B\cdot\mathrm d\mathbf a}}_{\Phi\equiv\, \textrm{magnetic flux}}\tag 1 \\ \underbrace{\int_\mathrm S\mathbf  B\cdot \mathrm d\mathbf a}_{\textrm{flux through}\,\color{red}{\textrm{closed surface}}} &= 0\tag 2\end{align}
OP can see the distinct red marked term in the first equation; the later doesn't talk like that.
The possible misinterpretation could be averted if OP transforms the differential form to integral form keeping in mind the Gauss' or Divergence theorem and the Stokes' Theorem.
