electromagnetism - 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?

Answer

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