Frequency domain(time-harmonic) maxwell equations

Frequency domain maxwell equation (vs. regular time-domain maxwell equations)
That's most frequently used maxwell equations. Every time domain function a(t) can be obtained using inverse fourier transform, A(w)

\begin{align*}
\pmb{e} & (\pmb{r},t)=\frac{1}{2\pi }\int \pmb{E}(\pmb{r},\omega)e^{j \omega t}d\omega,
\pmb{h}  (\pmb{r},t)=\frac{1}{2\pi }\int \pmb{H}(\pmb{r},\omega)e^{j \omega t}d\omega \\
F & or ~time ~ harmonic ~ components,\\
\mathcal{E} &(r,t)=\pmb{E}(r,\omega)e^{j\omega t},
\mathcal{H}(r,t)=\pmb{H}(r,\omega)e^{j\omega t}\\
\end{align*}

Substitute time harmonic E/H into maxwell equations, with d/dt=jw, we have frequency domain maxwell equations

\begin{align*}
\nabla & \times \pmb{E}= - j\omega \pmb{B} \\
\nabla & \times \pmb{H}= \pmb{J}+  j\omega \pmb{D}\\
\nabla & \cdot \pmb{D}= \rho\\
\nabla & \cdot \pmb{B}= 0
\end{align*}

Note: for the above equations, E=E(r,w). Often, we drop w for simplicity.
Once we know the solution of time harmonic E(r,w) and H(r,w) , we can obtain E(r,t) and H(r,t) in time domain by doing inverse fourier transform.

Signal phasor representation and time average
In real word, the signal are real. For a real value sinusodal signal,

\begin{align*}
 \mathcal{A} & (t)=|A|cos(\omega t+\phi) ~ \Longleftrightarrow  A(t)=Ae^{j\omega t}\\
o & r ~ \mathcal{A}(t)=Re \{A(t)\}\\
w&here ~ A=|A|e^{j\phi}~ is~ a ~complex ~phasor.
\end{align*}

Time average of two harmonic signals,

\begin{align*}
\mathcal{A} &(t)=Re [A(t)] = Re [Ae^{j\omega t} ]\\
\mathcal{B} &(t)=Re [B(t)] = Re [Be^{j\omega t} ]\\
<\mathcal{A} &(t) \mathcal{B}(t)>=<\frac{1}{2}(Ae^{j\omega t}+A^*e^{-j\omega t})*\frac{1}{2}(Be^{j\omega t}+B^*e^{-j\omega t})\\
= & \frac{1}{4}<(AB^*+BA^*)+ABe^{-j2\omega t}+A^*B^*e^{-j2\omega t}> \\
= & \frac{1}{2} ( < Re[AB^*] > + < Re[AB*e^{-j2\omega t} ] > ) \\
= & \frac{1}{2}  Re[AB^*]  
\end{align*}