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 $$\nabla \cdot \mathbf{J} + \dfrac{\partial}{\partial{t}}\rho = 0 \tag{2.1.5}$$ where no net generation or recombination of electrons is assumed. In the study of electromagnetic, one usually assumes that the source terms $\rho$ and $\mathbf{J}$ are given quantities. It is noted that (2.1.4) is derivable from (2.1.1) by taking the divergence of (2.1.1) and noting that $\nabla \cdot (\nabla \times \mathbf{E}) = 0$ for any vector $\mathbf{E}$.
So we are told that Gauss's law $\nabla \cdot \mathbf{B} = 0$ is derivable from Faraday's law $\nabla \times \mathbf{E} = - \dfrac{\partial}{\partial{t}}\mathbf{B}$ by taking the divergence and noting that $\nabla \cdot ( \nabla \times \mathbf{E}) = 0$ for any vector $\mathbf{E}$. I did not understand why/how $\nabla \cdot ( \nabla \times \mathbf{E}) = 0$, and nor did I understand how we then get $\nabla \cdot \mathbf{B} = 0$ from $\nabla \times \mathbf{E} = - \dfrac{\partial}{\partial{t}}\mathbf{B}$. So, since this seemed like a mathematical issue, I asked here. However, based on the comments, I am told that this actually isn't valid. So is the author just wrong here?
From the above point, the author then continues as follows:
Similarly, (2.1.3) is derivable from (2.1.2) using (2.1.5). Thus, we have only two independent vector equations (2.1.1) and (2.1.2), or six scalar equations as each vector has three components. However, there are $\mathbf{E}$, $\mathbf{H}$, $\mathbf{D}$, and $\mathbf{B}$, 12 scalar unknown components. Thus, we need six more scalar equations. These are the so-called constitutive relations that describe the properties of a medium. In isotropic media, they are given by $$\mathbf{D} = \epsilon \mathbf{E} \ \ \ \ \ \ \ \ \ \ \mathbf{B} = \mu \mathbf{H} \tag{2.1.6}$$ In anisotropic media, they may be given by $$\mathbf{D} = \epsilon \cdot \mathbf{E} \ \ \ \ \ \ \ \ \ \ \mathbf{B} = \mu \cdot \mathbf{H} \tag{2.1.7}$$ where $\epsilon$ is the permittivity tensor and $\mu$ is the permeability tensor: $$\epsilon = \begin{bmatrix} \epsilon_{xx} & \epsilon_{xy} & \epsilon_{xz} \\ \epsilon_{yx} & \epsilon_{yy} & \epsilon_{yz} \\ \epsilon_{zx} & \epsilon_{zy} & \epsilon_{zz} \end{bmatrix} \ \ \ \ \ \ \ \ \ \ \mu = \begin{bmatrix} \mu_{xx} & \mu_{xy} & \mu_{xz} \\ \mu_{yx} & \mu_{yy} & \mu_{yz} \\ \mu_{zx} & \mu_{zy} & \mu_{zz} \end{bmatrix} \tag{2.1.8}$$
For electromagnetic fields at optical frequencies, $\rho = 0$ and $\mathbf{J} = 0$.
Is all of this correct?
