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

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 $$\oint_{\partial \mathcal V} \mathbf {B} \cdot d\mathbf A=0$$