= Solution
The vector identity from Question 3 gives
$$
\begin{aligned}
\nabla\cdot(E\times B)
&=B\cdot(\nabla\times E)-E\cdot(\nabla\times B)\\
&=-B\cdot\frac{\partial B}{\partial t}
-E\cdot J-E\cdot\frac{\partial E}{\partial t}\\
&=-E\cdot J
-\frac{\partial}{\partial t}
\frac{|E|^2+|B|^2}{2}.
\end{aligned}
$$
Integrating over $V$ and applying the <divergence theorem> proves the <Poynting theorem>
$$
\boxed{
\int_SP\cdot dS
=-\int_VE\cdot J\,dV
-\frac{\partial}{\partial t}
\int_V\frac{|E|^2+|B|^2}{2}\,dV}.
$$
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