I am currently studying the textbook Physics of Photonic Devices, second edition, by Shun Lien Chuang. Section 2.1.1 Maxwell's Equations in MKS Units says the following:
The well-known Maxwell's equations in MKS (meter, kilogram, and second) units are written as $$\nabla \times \mathbf{E} = - \dfrac{\partial}{\partial{t}}\mathbf{B} \ \ \ \ \text{Faraday's law} \tag{2.1.1}$$ $$\nabla \times \mathbf{H} = \mathbf{J} + \dfrac{\partial{\mathbf{D}}}{\partial{t}} \ \ \ \ \text{Ampére's law} \tag{2.1.2}$$ $$\nabla \cdot \mathbf{D} = \rho \ \ \ \ \text{Gauss's law} \tag{2.1.3}$$ $$\nabla \cdot \mathbf{B} = 0 \ \ \ \ \text{Gauss's law} \tag{2.1.4}$$ where $\mathbf{E}$ is the electric field (V/m), $\mathbf{H}$ is the magnetic field (A/m), $\mathbf{D}$ is the electric displacement flux density (C/m$^2$), and $\mathbf{B}$ is the magnetic flux density (Vs/m$^2$ or Webers/m$^2$). The two source terms, the charge density $\rho$ (C/m$^3$) and the current density $\mathbf{J}$ (A/m$^2$), are related by the continuity equation $$\nabla \cdot \mathbf{J} + \dfrac{\partial}{\partial{t}} \rho = 0 \tag{2.1.5}$$
Section 2.1.2 Boundary Conditions then says the following:
By applying the first two Maxwell's equations over a small rectangular surface with a width $\delta$ (dashed line in Fig. 2.1a) across the interface of a boundary and using Stokes' theorem between a line integral over a contour $C$ and the surface $S$ enclosed by the contour $$\oint_C \mathbf{E} \cdot d \mathscr{l} = \int_S \nabla \times \mathbf{E} \cdot \mathbf{\hat{n}} \ dS = - \dfrac{d}{dt} \int_S \mathbf{B} \cdot \mathbf{\hat{n}} \ dS \tag{2.1.9a}$$ $$\oint_C \mathbf{H} \cdot d \mathscr{l} = \int_S \nabla \times \mathbf{H} \cdot \mathbf{\hat{n}} \ dS = \int_S \mathbf{J} \cdot \mathbf{\hat{n}} \ dS + \dfrac{d}{dt} \int_S \mathbf{D} \cdot \mathbf{\hat{n}} \ dS, \tag{2.1.9b}$$ the following boundary conditions can be derived by letting the width $\delta$ approach zero: $$\mathbf{\hat{n}} \times (\mathbf{E}_1 - \mathbf{E}_2) = 0 \tag{2.1.10}$$ $$\mathbf{\hat{n}} \times (\mathbf{H}_1 - \mathbf{H}_2) = \mathbf{J}_s, \tag{2.1.11}$$ where $\mathbf{J}_s(= \lim\limits_{\mathbf{J} \to \infty, \ \delta \to 0} \mathbf{J} \delta)$ is the surface current density (A/m). Note that the unit normal vector $\hat{n}$ points from medium 2 to medium 1.
How does letting $\delta$ approach zero get us 2.1.10 and 2.1.11? And why do we have $\mathbf{E}_1 - \mathbf{E}_2$ and $\mathbf{H}_1 - \mathbf{H}_2$ instead of $\mathbf{E}_2 - \mathbf{E}_1$ and $\mathbf{H}_2 - \mathbf{H}_1$?