A strange formulation of the Maxwell's equations

Question

I am working on a project that involves Maxwell's equations, and the research paper that I am working on has formulated the equations in a strange and unfamiliar way. Here are the four equations (1)-(4):

$\oint_{\partial A} \mathbf{E} \cdot \mathrm{d}\boldsymbol{s} = -\iint_{A} \frac{\partial \mathbf{B}}{\partial t} \cdot \mathrm{d}\mathbf{A}$

$\oint_{\partial A} \mathbf{H} \cdot \mathrm{d}\boldsymbol{s} = \iint_{A} \left(\frac{\partial \mathbf{D}}{\partial t} + \mathbf{J}\right) \cdot \mathrm{d}\mathbf{A}$

$\iint_{\partial \mathbf{V}} \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0$

$\iint_{\partial \mathbf{V}} \left(\frac{\partial \mathbf{D}}{\partial t} + \mathbf{J}\right) \cdot \mathrm{d}\mathbf{A} = 0$

With the following relation equations (5)-(7):

$D=\epsilon E$, $B=\mu H$, and $J=\kappa E + \rho v$

I got the first 3 equation, which are respectively Maxwell-Faraday, Maxwell Ampere and Guass's law for magnetism. What I do not get is the supposed Gauss's law (4th equation), which does not look like a Gauss'S law to me as well as the last relation equation relating the current density with the E field and the movement of the particles inside the medium. Mind someone shedding some light over the two? I would be extremely thankful since I have been pulling my hair to understand them since two weeks.

The paper mentioning the formulation can be found here: https://doi.org/10.1016/0010-4655(92)90026-U

Answer 1

This is a strange notation. You have for any $\mathbf F$ vector field $\int_{\mathcal V} dV \text{div} \mathbf F = \oint_{\partial \mathcal V} \mathbf F \cdot d\mathbf A$ , where $\oint$ is a surface integral and ${\partial \mathcal V}$ is the closed boundary surface of the volume ${\mathcal V}$.

Apply this to $\mathbf {curl H}=\mathbf {J + \dot D}$ and note that $\text{div}\mathbf {curl}=0$, you get $$\oint_{\partial \mathcal V} (\mathbf {J + \dot D}) \cdot d\mathbf A=0$$

The same argument holds for Eq.3 where $\text{div}\mathbf B = 0$, thus $$\oint_{\partial \mathcal V} \mathbf {B} \cdot d\mathbf A=0$$

Answer 2

The usual $\mathrm{div}~{\bf B}=0$ and $\mathrm{div}~{\bf D}=\rho$ equations are not dynamical equations (they do not contain time derivatives, see for example, this question). Thus, they can be considered as a constraint on the initial condition, provided their time derivative is zero. Indeed, the vanishing of the time derivatives is ensured by the validity of the curl equations and the continuity equation for the current. In an integral formulation, such a condition must be explicitly present.

The last equation is nothing but the condition that $$ \frac{\mathrm{d}}{\mathrm{d}t} \int_{V} \left(\mathrm{div}~\mathbf{D} - \rho\right) \cdot \mathrm{d}V =0 $$