I am currently studying Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th edition, by Max Born and Emil Wolf. Chapter 1.1.1 Maxwell's equations says the following:
The space and times derivatives of the five vectors are related by Maxwell's equations, which hold at every point in whose neighbourhood the physical properties of the medium are continuous: $$\text{curl} \ \mathbf{H} - \dfrac{1}{c} \mathbf{\dot{D}} = \dfrac{4\pi}{c} \mathbf{j}, \tag{1}$$ $$\text{curl} \ \mathbf{E} + \dfrac{1}{c} \mathbf{\dot{B}} = 0, \tag{2}$$ the dot denoting differentiation with respect to time. They are supplemented by two scalar relations: $$\text{div} \ \mathbf{D} = 4 \pi \rho, \tag{3}$$ $$\text{div} \ \mathbf{B} = 0 . \tag{4}$$ Eq. (3) may be regarded as a defining equation for the electric charge density $\rho$ and (4) may be said to imply that no free magnetic poles exist.
From (1) it follows (since $\text{div} \ \text{curl} \ \equiv 0$) that $$\text{div} \ \mathbf{j} = - \dfrac{1}{4\pi} \text{div} \ \mathbf{\dot{D}},$$ or, using (3), $$\dfrac{\partial{\rho}}{\partial{t}} + \text{div} \ \mathbf{j} = 0. \tag{5}$$ By analogy with a similar relation encountered in hydrodynamics, (5) is called the equation of continuity. It expresses the fact that the charge is conserved in the neighbourhood of any point, for if one integrates (5) over any region of space, one obtains, with the help of Gauss' theorem, $$\dfrac{d}{dt} \int \rho \ dV + \int \mathbf{j} \cdot \mathbf{n} \ dS = 0, \tag{6}$$ the second integral being taken over the surface bounding the region and the first throughout the volume, $\mathbf{n}$ denoting the unit outward normal.
I have a number of questions about going from (5) to (6). Taking the integral of (5), we get
$$\int \dfrac{d \rho}{dt} + \text{div} \ \mathbf{j} \ dV = 0 \\ \Rightarrow \int \dfrac{d \rho}{dt} \ dV + \int \text{div} \ \mathbf{j} \ dV = 0$$
Apparently, we then apply Gauss's theorem (which is equivalent to Green's theorem in 2 dimensions), to the second term. But, according to the theorem, we are supposed to go from the $n$-dimensional volume integral to the ($n - 1$)-dimensional surface integral, so I don't understand how it is applied in this case. Furthermore, I think I remember from studying analysis that we cannot just interchange the integral and partial derivatives to get $\dfrac{d}{dt} \int \rho$ from $\int \dfrac{d \rho}{dt}$ without satisfying some conditions. I would greatly appreciate it if people would please take the time to clarify all of this.