Divergence of Et in TEM propagation in transmission lines
For example, according to 7.1.37 $$\text{div}\textbf{H}=\text{div}_T\textbf{H}_T+\frac{\partial}{\partial z}\hat {\textbf{z}} \cdot \textbf{H}_T $$
but you know that there are no magnetic space charges $\text{div}\textbf{H}=0$ and by definition $\hat {\textbf{z}} \cdot \textbf{H}_T$, therefore you must also have 
$$\text{div}_T\textbf{H}_T=0$$; same consideration for the $\textbf{E}$ field.

The induced surface charges or currents play no role in the spatial divergence of the fields but they are important to define the boundary conditions: the E-field is perpendicular and $\propto  \sigma$ and H-field is tangential with the ideal metal surface and $\propto  \K$.