The Maxwell's equations are

\begin{align}
\frac{\partial \vec{B}}{\partial t} &= - \nabla \times \vec{E}            \hspace{2cm} \text{Faraday equations}\\
\frac{\partial \vec{D}}{\partial t} &= - \nabla \times \vec{H} - \vec{J} - \vec{s} \hspace{2cm} \text{Ampere's law.}
\end{align}

where $\vec{E}$ is the electric field, $\vec{H}$ is the magnetic field, $\vec{J}$ is the electric flux, $\vec{D}$ is the displacement flux, $\vec{B}$ is the magnetic flux and $\vec{s}$ is the source.

The constitutive realtations between the fields and flux are as follows.

\begin{align}
\vec{J} &= \sigma \vec{E}\\
\vec{B} &= \mu \vec{H}\\
\vec{D} &= \epsilon \vec{E}
\end{align}

Using the contitutive relations, the Maxwell's equations can written in terms of the electric field and the magnetic flux 
\begin{align}
\mu \frac{\partial \vec{B}}{\partial t} &= - \nabla \times \vec{E}  \\
\epsilon \frac{\partial \vec{E}}{\partial t} &= -  \nabla \times \frac{1}{\mu} \vec{B} - \sigma \vec{E} - \vec{s} 
\end{align}

Futher by droping the $\epsilon$ term and using a $e^{-i\omega t}$ Fourier time relation, Maxwell's equations can be written in the frequency domain as

\begin{align}
-i \omega \mu  \hat{\vec{B}} &= - \nabla \times \hat{\vec{E}}  \\
- \vec{s} &= -  \nabla \times \frac{1}{\mu} \hat{\vec{B}} - \sigma \hat{\vec{E}}
\end{align}





The weak form is 
\begin{align}
-i \omega \mu (\hat{\vec{B}},\hat{\vec{F}}) + (\nabla \times \hat{\vec{E}},\hat{\vec{F}}) &= 0 \\
(\nabla \times \mu^{-1} \hat{\vec{B}},\hat{\vec{W}}) + (\sigma \hat{\vec{E}},\hat{\vec{W}}) &= (\hat{\vec{s}},\hat{\vec{W}})
\end{align}


$ (\mu^{-1} B, \nabla \times W) + (\sigma E, W) = (\mu^{-1}B W )|_0^{end} $

$ (i \omega B,F) + (\nabla \times E , f) = 0 $


```

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```

```

d94031f6afaeb388c8c988a51b1cbad5332b5cbf
ipynb
Jupyter Notebook
notebooks/3D theroy overview-Copy0.ipynb
hanbo1735/simpegmt
73e35c4944c3e14a0433fbd7c3f01cae2fc11ac6
MIT
2015-03-31T23:21:16.000Z
2021-07-26T11:14:31.000Z
notebooks/3D theroy overview-Copy0.ipynb
hanbo1735/simpegmt
73e35c4944c3e14a0433fbd7c3f01cae2fc11ac6
MIT
2015-05-04T18:17:29.000Z
2015-12-23T00:11:51.000Z
notebooks/3D theroy overview.ipynb
simpeg/simpegmt
73e35c4944c3e14a0433fbd7c3f01cae2fc11ac6
MIT
2015-03-31T23:21:18.000Z
2021-10-06T12:35:44.000Z
Qwen/Qwen-72B
1. YES
2. YES