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3.1: Poynting’s Theorem
https://phys.libretexts.org/Bookshelves/Electricity_and_Magnetism/Electromagnetics_II_(Ellingson)/03%3A_Wave_Propagation_in_General_Media/3.01%3A_Poyntings_Theorem
\[\begin{aligned} - \int_{\mathcal{V}} &\nabla \cdot \left( {\bf E} \times {\bf H} \right) dv \nonumber \\ &= \int_{\mathcal{V}} {\bf E} \cdot {\bf J}~dv \nonumber \\ &+ \int_{\mathcal{V}} \frac{1}{2}... $\begin{aligned} - \int_{\mathcal{V}} &\nabla \cdot \left( {\bf E} \times {\bf H} \right) dv \nonumber \\ &= \int_{\mathcal{V}} {\bf E} \cdot {\bf J}~dv \nonumber \\ &+ \int_{\mathcal{V}} \frac{1}{2} \frac{\partial}{\partial t} \left( {\bf E}\cdot{\bf D} \right) dv \nonumber \\ &+ \int_{\mathcal{V}} \frac{1}{2} \frac{\partial}{\partial t} \left( {\bf H}\cdot{\bf B} \right) dv\end{aligned} \nonumber$

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