\partial^\mu = \left(\begin{matrix}
\frac{1}{c}\partial_t\
-\partial_x\
-\partial_y\
-\partial_z
\end{matrix}\right)
::::
A^\mu = \left(\begin{matrix}
\phi \
A_x \
A_y \
A_z
\end{matrix}\right)
::::
g_{\mu\nu}=\left(\begin{matrix} 1 & 0 & 0 & 0\ 0 & -1 & 0 & 0\ 0 & 0 & -1 & 0\ 0 & 0 & 0 & -1 \end{matrix}\right)
\varepsilon^{\mu\nu\sigma\tau} = \begin{cases}
& +1 \text{ bei geraden Permutationen von 0123 }\
& -1 \text{ bei ungeraden Permutationen von 0123 }\
& 0 \text{ sonst }
\end{cases}
F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu
\color{red}F^{\mu\nu} = \left(\begin{matrix}
0 &:: \frac{1}{c}\partial_t A_x + \partial_x \phi :: & :: \frac{1}{c}\partial_t A_y + \partial_y \phi :: & ::\frac{1}{c}\partial_t A_z + \partial_z \phi\
-\partial_x \phi - \frac{1}{c}\partial_t A_x ::& 0 & \partial_y A_x -\partial_x A_y& \partial_z A_x -\partial_x A_z \
-\partial_y \phi - \frac{1}{c}\partial_t A_y ::& \partial_x A_y -\partial_y A_x & 0 & \partial_z A_y-\partial_y A_z \
-\partial_z \phi - \frac{1}{c}\partial_t A_z ::& \partial_x A_z -\partial_z A_x & \partial_y A_z -\partial_z A_y & 0
\end{matrix}\right)
F_{\mu\nu} = g_{\mu\alpha} g_{\nu\beta} F^{\alpha\beta}
F_{\mu\nu}=\left(\begin{matrix}
0 & ::-\partial_x \phi - \frac{1}{c}\partial_t A_x ::&:: -\partial_y \phi - \frac{1}{c}\partial_t A_y ::&:: -\partial_z \phi -\frac{1}{c}\partial_t A_z::\
\frac{1}{c}\partial_t A_x + \partial_x \phi& 0 ::& \partial_y A_x -\partial_x A_y & \partial_z A_x -\partial_x A_z \
\frac{1}{c}\partial_t A_y +\partial_y \phi & \partial_x A_y -\partial_y A_x & 0 & \partial_z A_y-\partial_y A_z \
\frac{1}{c}\partial_t A_z + \partial_z \phi& \partial_x A_z +\partial_z A_x & \partial_y A_z -\partial_z A_y & 0
\end{matrix}\right)
\widehat{F}^{\mu\nu} = \frac{1}{2}\varepsilon ^{\mu\nu\sigma\tau} F_{\sigma\tau}
\widehat{F}^{\mu\nu} = \frac{1}{2}\left(\begin{matrix} 0 & ::F_{23} - F_{32}:: & ::F_{31} - F_{13}:: & ::F_{12} - F_{21}::\ F_{32} - F_{23} & 0 & F_{03} - F_{30} & F_{20} - F_{02}\ F_{13} - F_{31} & F_{30} - F_{03} & 0 & F_{01} - F_{10}\ ::F_{21} - F_{12}:: & F_{02} - F_{20} & F_{10} - F_{01} & 0 \end{matrix}\right)
\color{red}\widehat{F}^{\mu\nu}=\left(\begin{matrix} 0 & :: \partial_z A_y-\partial_y A_z :: & :: \partial_x A_z -\partial_z A_x :: & :: \partial_y A_x -\partial_x A_y ::\ :: \partial_y A_z -\partial_z A_y :: & 0 & -\partial_z \phi -\frac{1}{c}\partial_t A_z & \partial_y \phi + \frac{1}{c}\partial_t A_y \ \partial_z A_x-\partial_x A_z & \partial_z \phi +\frac{1}{c}\partial_t A_z & 0 & -\partial_x \phi - \frac{1}{c}\partial_t A_x \ \partial_x A_y -\partial_y A_x & -\partial_y \phi - \frac{1}{c}\partial_t A_y & \partial_x \phi +\frac{1}{c}\partial_t A_x & 0 \end{matrix}\right)
\color{red}H^{\mu\nu}=\left\langle F^{\mu\nu}{frei} \right\rangle + \widehat{F}^{\mu\nu}{hilf} ::: , :::
4\pi M^{\mu\nu}=\left\langle F^{\mu\nu}{mat} \right\rangle - \widehat{F}^{\mu\nu}{hilf}
F^{\mu\nu}=F^{\mu\nu}(\vec{E},\vec{B}) ::: , ::: H^{\mu\nu}=H^{\mu\nu}(\vec{D},\vec{H}) ::: , ::: M^{\mu\nu}=M^{\mu\nu}(-\vec{P},\vec{M})
E_i = F^{i0}=\left(\begin{matrix} -\partial_x \phi - \frac{1}{c}\partial_t A_x \
-\partial_y \phi - \frac{1}{c}\partial_t A_y \
-\partial_z \phi - \frac{1}{c}\partial_t A_z
\end{matrix}\right) = -\vec{\nabla}\phi - \frac{1}{c}\frac{\partial}{\partial t}\vec{A}= \left(\begin{matrix} E_x \ E_y \ E_z \end{matrix}\right) = \vec{E}
\color{red}D_i = \left\langle F^{i0}{frei}\right\rangle + \left\langle\widehat{F}^{i0}\right\rangle = \left\langle\vec{E}{frei}\right\rangle + rot \vec{A}
\color{red}-4\pi P_i = \left\langle F^{i0}{mat}\right\rangle + \left\langle\widehat{F}^{i0}\right\rangle = \left\langle\vec{E}{mat}\right\rangle + rot \vec{A}
-\varepsilon_{ijk}B_k = F^{ij} = \left(\begin{matrix}
0 &:: \partial_y A_x -\partial_x A_y ::&:: \partial_z A_x -\partial_x A_z \
\partial_x A_y -\partial_y A_x ::& 0 & \partial_z A_y-\partial_y A_z \
\partial_x A_z +\partial_z A_x & \partial_y A_z -\partial_z A_y & 0
\end{matrix}\right)=\left(\begin{matrix}0 & -B_z & B_y \
B_z & 0 & -B_x \ -B_y & B_x & 0\end{matrix}\right)
-\varepsilon_{ijk}H_k = \left\langle F_{frei}^{ij}\right\rangle + \widehat{F}_{hilf}^{ij}= \left(\begin{matrix}0 & -B_z & B_y \
B_z & 0 & -B_x \ -B_y & B_x & 0\end{matrix}\right) +
\left(\begin{matrix}0 &:: -\partial_z \phi -\frac{1}{c}\partial_t A_z ::&:: \partial_y \phi +\frac{1}{c}\partial_t A_y \
\partial_z \phi + \frac{1}{c}\partial_t A_z :: & 0 & - \partial_x \phi - \frac{1}{c}\partial_t A_x \
- \partial_y \phi - \frac{1}{c}\partial_t A_y & \partial_x \phi + \frac{1}{c}\partial_t A_x & 0\end{matrix}\right)
-\varepsilon_{ijk}4\pi M_k = \left\langle F_{mat}^{ij}\right\rangle - \widehat{F}_{hilf}^{ij}
\color{red}\vec{H} = \left\langle \vec{B}_{frei} \right\rangle + \vec\nabla\phi + \frac{1}{c}\frac{\partial}{\partial}t \vec{A}
\color{red}4\pi\vec{M} = \left\langle \vec{B}_{mat} \right\rangle - \vec\nabla\phi - \frac{1}{c}\frac{\partial}{\partial}t \vec{A}