= Solution
Allow time dependence while keeping the <constitutive relations> linear, isotropic, nondispersive, and independent of time. The <Maxwell equations in matter> give
$$
\nabla\times\mathbf E=-\partial_t\mathbf B,\qquad
\nabla\times\mathbf H=\mathbf J_f+\partial_t\mathbf D.
$$
For the <Poynting vector> $\boldsymbol{\mathcal P}=\mathbf E\times\mathbf H$, the <divergence> identity yields
$$
\begin{aligned}
\nabla\cdot\boldsymbol{\mathcal P}
&=\mathbf H\cdot(\nabla\times\mathbf E)-\mathbf E\cdot(\nabla\times\mathbf H)\\
&=-\frac1\mu\mathbf B\cdot\partial_t\mathbf B
-\epsilon\mathbf E\cdot\partial_t\mathbf E-\mathbf E\cdot\mathbf J_f.
\end{aligned}
$$
The first two terms are exact time <derivatives>. Taking the zero-field <energy density> as zero gives the <electromagnetic energy density>
$$
\boxed{\mathcal E=\frac12\epsilon|\mathbf E|^2+\frac1{2\mu}|\mathbf B|^2
=\frac12(\mathbf E\cdot\mathbf D+\mathbf B\cdot\mathbf H).}
$$
Consequently the <Poynting theorem> is the <continuity equation>
$$
\boxed{\partial_t\mathcal E+\nabla\cdot\boldsymbol{\mathcal P}=-\mathbf J_f\cdot\mathbf E.}
$$
In integral form,
$$
\frac d{dt}\int_W\mathcal E\,d^3r+
\oint_{\partial W}\boldsymbol{\mathcal P}\cdot\mathbf n\,dS
=-\int_W\mathbf J_f\cdot\mathbf E\,d^3r.
$$
The left side is the rate of increase of stored <electromagnetic energy> plus outward <Poynting flux>. The right side accounts for work delivered to the free <electric current>; in an Ohmic medium $\mathbf J_f\cdot\mathbf E$ is the local rate of <Joule heating>.
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