-i\mu_r \hat{\mathbf{H}} + \hat{\nabla} \times \hat{\mathbf{E}}
&= -\hat{\mathbf{M}}_a,
\\
- \hat{\nabla} \cdot (\mu_r\hat{\mathbf{H}}) &= \frac{1}{i\omega}\hat{\nabla}
+ \hat{\nabla} \cdot (\mu_r\hat{\mathbf{H}}) &= \frac{1}{i}\hat{\nabla}
\cdot \hat{\mathbf{M}}_a,
\\
i\varepsilon_r\hat{\mathbf{E}} + \nabla\times(\mu^{-1}\mathbf{H})
&= \mathbf{J}_a,
\\
- \nabla\cdot(\varepsilon\mathbf{E}) &= \frac{1}{i\omega}\hat{\nabla}
+ \nabla\cdot(\varepsilon\mathbf{E}) &= \frac{1}{i}\hat{\nabla}
\cdot\hat{\mathbf{J}}_a.
@f}
- \int_\Omega \varepsilon_r\mathbf{E} \cdot \bar{\varphi}\;\text{d}x
- \int_\Sigma [\nu \times (\mu_r^{-1}\nabla\times\mathbf{E} +
\mu^{-1}\mathbf{M}_a)]_{\Sigma}\cdot \bar{\varphi}_T\;\text{d}o_x\\
-\qquad - \int_{\partial\Omega} (\nu \times (\mu_r^{-1}\nabla\times\mathbf{E} +
+\qquad + \int_{\partial\Omega} (\nu \times (\mu_r^{-1}\nabla\times\mathbf{E} +
\mu^{-1}\mathbf{M}_a)) \cdot \bar{\varphi}_T\;\text{d}o_x =
i\int_\Omega \mathbf{J}_a \cdot \bar{\varphi}\;\text{d}x
- \int_\Omega \mu_r^{-1}\mathbf{M}_a \cdot (\nabla \times \bar{\varphi})\;\text{d}x.
- \int_\Omega \varepsilon_r\mathbf{E} \cdot \bar{\varphi}\;\text{d}x
- i\int_\Sigma (\sigma_r^{\Sigma}\mathbf{E}_T) \cdot \bar{\varphi}_T\;\text{d}o_x\\
\qquad - i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}\mathbf{E}_T) \cdot
-(\nabla\times\bar{\varphi}_T)\;\text{d}o_x.=
+\bar{\varphi}_T\;\text{d}o_x =
i\int_\Omega \mathbf{J}_a \cdot \bar{\varphi}\;\text{d}x
- \int_\Omega \mu_r^{-1}\mathbf{M}_a \cdot (\nabla \times \bar{\varphi})\;\text{d}x.
@f]
- \int_\Omega \varepsilon_r\mathbf{E} \cdot \bar{\varphi}\;\text{d}x
- i\int_\Sigma (\sigma_r^{\Sigma}\mathbf{E}_T) \cdot \bar{\varphi}_T\;\text{d}o_x
- i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}\mathbf{E}_T) \cdot
-(\nabla\times\bar{\varphi}_T)\;\text{d}o_x.\\
+\bar{\varphi}_T\;\text{d}o_x.\\
F(\varphi) \dealcoloneq i\int_\Omega \mathbf{J}_a \cdot \bar{\varphi}\;\text{d}x
- \int_\Omega \mu_r^{-1}\mathbf{M}_a \cdot (\nabla \times \bar{\varphi})\;\text{d}x.
@f]
- i\int_\Sigma (\sigma_r^{\Sigma}\varphi_{j_T}) \cdot
\bar{\varphi}_{i_T}\;\text{d}o_x
- i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}\varphi_{j_T})
- \cdot (\nabla\times \bar{\varphi}_{i_T})\;\text{d}o_x,
+ \cdot \bar{\varphi}_{i_T}\;\text{d}o_x,
@f]
@f[
F_i = i\int_\Omega \mathbf{J}_a \cdot \bar{\varphi_i}\;\text{d}x