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The question is supposed to read "evaluate all five terms in the equation (6)"

The textbook derives following expression for Poynting's theorem in case of linear dispersive medium (page 264): \[ \frac{\partial u_{e \ddot{f}}}{\partial t}+\vec{\nabla} \cdot \vec{S}=-\vec{J} \cdot \vec{E}-2 \omega_{0} \operatorname{Im} \varepsilon\left(\omega_{0}\right)\langle\vec{E}(t, \vec{r}) \cdot \vec{E}(t, \vec{r})\rangle-2 \omega_{0} \operatorname{Im} \mu\left(\omega_{0}\right)\langle\vec{H}(t, \vec{r}) \cdot \vec{H}(t, \vec{r})\rangle, \] where the effective electromagnetic energy density is defined as, \[ u_{e f f}=\operatorname{Re}\left[\frac{d(\omega \varepsilon)}{d \omega}\right]_{\omega=\omega_{0}}\langle\vec{E}(t, \vec{r}) \cdot \vec{E}(t, \vec{r})\rangle+\operatorname{Re}\left[\frac{d(\omega \mu)}{d \omega}\right]_{\omega=\omega_{b}}\langle\vec{H}(t, \vec{r}) \cdot \vec{H}(t, \vec{r})\rangle . \] Assuming that \( E \)-field solution for Maxwell equations is given by real part of (4), evaluate all five terms in the eq. (5); Explain what is physical meaning of these five terms; Do the lefthand side and the right-hand side of (5) match? Hint: When calculating the pointing vector, you may apply averaging over one oscillation period. A mathematically more rigorous averaging procedure for oscillatory functions will be discusses later on.
\( \vec{E}(z, t)=\vec{e}_{x} E_{0} e^{-\frac{1}{2} \alpha(\omega) z} e^{-i \omega\left(t \frac{n(\omega) z}{c}\right)} \)

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