Maxwell’s first equation or Gauss’s law in electrostatics
Statement. It states that the total electric flux φE passing through a closed hypothetical surface is equal to 1/ε0 times the net charge enclosed by the surface:
where D=ε0E= Displacement vector
Let the charge be distributed over a volume V and p be the volume charge density .therefore q=∫ pdV
Therefore ∫ D.dS=∫vpdV (1)
Equation (1) is the integral form of Maxwell’s first equation or Gauss’s law in electrostatics.
Apply Gauss’s Divergence theorem to change L.H.S. of equation(1) from surface integral to volume integral
That is ∫ D.dS=∫( ∇.D)dV
Substituting this equation in equation (1), we get
∫ (∇.D)dV=∫v pdV
As two volume integrals are equal only if their integrands are equal
Thus, ∇.D=p (2)
Equation (2) is the Differential form of Maxwell’s first equation.
Maxwell’s second equation or Gauss’s law for Magnetism
Statement. It states that the total magnetic flux φm emerging through a closed surface is zero.
The equation (3) is the Intergal form of Maxwell’s second equation.
This equation also proves that magnetic monopole does not exist.
Apply Gauss’s Divergence theorem to equation (3)
That is ∫s B.dS=∫v(∇.B)dV
Thus , ∇ .B=0 (4)
The equation (4) is differential form of Maxwell’s second equation.
Note: You can also read article on Maxwell third equation and its derivation.