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Maxwell first equation

The Maxwell first equation in electrostatics is called the Gauss law in electrostatics.

Statement: 
It states that the total electric flux \(\psi_E\) passing through a closed hypothetical surface is equal to
\(\frac{1}{\epsilon_0}\)
enclosed by the surface.

Integral Form:
$$\phi_E = \int E.ds = \frac{q}{\epsilon_0}$$
$$\int D.ds = q$$
where,
$$D = \epsilon_0 E = displacement-vector$$
Let the change be distributed over a volume v and \(\rho\) be the volume charge density.
Hence,
$$q = \int \rho dv$$
Therefore;
$$\int D.ds = \int_{v} \rho dv$$ .........(1)

Equ(1) is the integral form of Maxwell first law

Differential form:
Apply Gauss divergence theorem to the L.H.S of equ(1) from surface integral to volume integral.
$$\int D.ds = \int (\nabla.D)dv$$
Substituting this equation to equ(1)
$$\int(\nabla.D)dv = \int_{v} \rho dv$$
As two volume integrals are equal only if their integrands are equal.
Thus;
$$\nabla.D = \rho v$$ ............(2)

Equ(2) is the differential form of Maxwell first equation.
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