Divergence of non conservative electric field I'm looking for the proof that the 1st Maxwell equation is valid also on non conservative electric field.
When we are talking about a electrostatic field, the equation is ok. We can apply the Gauss (or Flux) theorem and get Gauss' law:
$$\mathbf{\nabla} \cdot \mathbf{E} ~=~ \frac{1}{\epsilon _0} \rho (x,y,z).$$
The question is, why when there is a time dependent magnetic field, and then a time dependent (non conservative) induced electric field, the 1st Maxwell equation is the same?
How we can prove that?
 A: As you've said, and just to be completely clear, in vacuum (neglecting, in other words, effects in macroscopic media like polarization), Gauss' law is the full, time-dependent expression of what you're calling the "first Maxwell equation."
The "derivation" of the Maxwell equations were originally formulated as differential (local) versions of the well known empirically observed laws of Ampere, Faraday, and Gauss. This is discussed some in Jackson's book ("Classical Electrodynamics"). Also see Griffith's book ("Intro to Electrodynamics"). 
The Maxwell equations aren't really derived from more fundamental considerations. Their integral form (the "laws" cited above) were deduced from observation, compared with phenomena not originally used in the determination of the empirical "laws," and found, in some regimes, to work. 
In the regime of atomic physics, Planck found that the assumed continuous radiation of an accelerating charge predicted a black-body spectrum at large frequency in contradiction with that observed. And this led to a modification of the classical electrodynamics and the advent of the quantum theory.
The form of the Maxwell equations is, however, tightly constrained by invariance under Lorentz transformations. Jackson discusses this in Chapter 11.
A: Great question. Almost all authors don't show that further justification is is needed to get Gaus's law for induced(time dependent) electric fields
The third Hertz’s equation for electrostatic field is a generalization of the Gauss law for electrostatic fields arrived at as follows:
$$\nabla \cdot E_{static} = \frac{Q}{e}$$  - Gaus's law for electrostatic(time independent) fields.
From the fact that induced electric field does not have sources(think of Faradays coil induction experiment with a centrally placed straight round core in the induction coil to avoid distracting asymmetries. The induced E field is radially symmetric - another way it is usually stated is: the induced EMF is distributed), it follows at once that
$$\nabla \cdot E_{induced} = 0$$ (Use the divergence theorem to convince yourself)
Summing these two equations, we get the differential form of the third Maxwell-Hertz’s equation: 
$$\nabla \cdot E_{total} = \nabla \cdot [E_{static} + E_{induced}] =  0$$ 
in the absence of charges
