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 1.1.3 Boundary conditions at a surface of discontinuity says the following:
Let us replace the sharp discontinuity surface $T$ by a thin transition layer within which $\epsilon$ and $\mu$ vary rapidly but continuously from their values near $T$ on one side to their value near $T$ on the other. Within this layer we construct a small near-cylinder, bounded by a stockade of normals to $T$; roofed and floored by small area $\delta A_1$ and $\delta A_2$ on each side of $T$, at constant distance from it, measured along their common normal (Fig. 1.1). Since $\mathbf{B}$ and its derivatives may be assumed to be continuous throughout this cylinder, we may apply Gauss's theorem to the integral of $\text{div} \ \mathbf{B}$ taken throughout the volume of the cylinder and obtain, from (4), $$\int \text{div} \ \mathbf{B} \ dV = \int \mathbf{B} \cdot \mathbf{n} \ dS = 0; \tag{12}$$ the second integral is taken over the surface of the cylinder, and $\mathbf{n}$ is the unit outward normal. Since the area $\delta A_1$ and $\delta A_2$ are assumed to be small, $\mathbf{B}$ may be considered to have constants values $\mathbf{B}^{(1)}$ and $\mathbf{B}^{(2)}$ on $\delta A_1$ and $\delta A_2$, and (12) may then be replaced by $$\mathbf{B}^{(1)} \cdot \mathbf{n}_1 \delta A_1 + \mathbf{B}^{(2)} \cdot \mathbf{n}_2 \delta A_2 + \text{contribution from walls} = 0 \tag{13}$$
$\epsilon$ is the dielectric constant (or permittivity).
$\mu$ is the magnetic permeability.
$\mathbf{B}$ is the magnetic induction.
(4) is the relation $\text{div} \ \mathbf{B} = 0$.
Why does the area $\delta A_1$ and $\delta A_2$ being small imply that $\mathbf{B}$ may be considered to have constants values $\mathbf{B}^{(1)}$ and $\mathbf{B}^{(2)}$ on $\delta A_1$ and $\delta A_2$? It almost seems like the terms $\mathbf{B}^{(k)} \cdot \mathbf{n}_k \delta A_k$ are some height times area calculation.
I would greatly appreciate it if people would please take the time to clarify this.