The T.M. Field

The T.M. has its source and vector potential four-vectors lie strictly in the 2-d Lorentz plane:

$$( S_{\tau}, S_{\xi}, S_{r}, S_{\theta})=\left( \xi\frac{\par... ...tial \tau},0,0 \right)\quad \textrm{in }I\textrm{ or in }II~,$$ (31)

$$( A_{\tau}, A_{\xi}, A_{r}, A_{\theta})=\left( \xi\frac{\par... ...rtial \tau},0,0\right)\quad \textrm{in }I\textrm{ or in }II~,$$ (32)

and

$$( A_{\xi}, A_{\tau}, A_{r}, A_{\theta})=\left( \frac{1}{\xi}... ...\partial \xi},0,0\right)\quad \textrm{in }F\textrm{ or in }P~$$ (33)

The components of the T.M. Maxwell field are

	In $I$ or in $II$	In $F$ or in $P$
$\hat E_{long.}:$	$$\frac{1}{\xi}F_{\xi\tau}= \left( \frac{1}{\xi}\frac{\partial}{\... ...}{\partial \xi} -\frac{1}{\xi^2}\frac{\partial^2}{\partial\tau^2} \right) \psi$$	$$\frac{1}{\xi}F_{\tau\xi}= -\left( \frac{1}{\xi}\frac{\partial}{\... ...{\partial \xi} -\frac{1}{\xi^2}\frac{\partial^2}{\partial\tau^2} \right) \psi$$
$\hat E_r:$	$$\frac{1}{\xi}F_{r\tau}= \frac{\partial}{\partial r}\left( \frac{\partial \psi}{\partial \xi} \right)$$	$$F_{r\xi}= \frac{\partial}{\partial r}\left( \frac{1}{\xi}\frac{\partial \psi}{\partial \tau} \right)$$
$\hat E_\theta:$	$$\frac{1}{\xi r}F_{\theta\tau}= \frac{1}{r} \frac{\partial}{\partial \theta} \left( \frac{\partial \psi}{\partial \xi} \right)$$	$$\frac{1}{r}F_{\theta\xi}= \frac{1}{r} \frac{\partial}{\partial \theta}\left ( \frac{1}{\xi}\frac{\partial \psi}{\partial \tau} \right)$$
$\hat B_{long.}:$	$$\frac{1}{r}F_{r\theta}=0$$	$$\frac{1}{r}F_{r\theta}=0$$
$\hat B_r:$	$$\frac{1}{r}F_{\theta\xi}= \frac{1}{r} \frac{\partial}{\partial \theta}\left ( \frac{1}{\xi}\frac{\partial \psi}{\partial \tau} \right)$$	$$\frac{1}{\xi r}F_{\theta\tau}= \frac{1}{r} \frac{\partial}{\partial \theta} \left( \frac{\partial \psi}{\partial \xi} \right)$$
$\hat B_\theta:$	$$F_{\xi r}= -\frac{\partial}{\partial r} \left( \frac{1}{\xi}\frac{\partial \psi}{\partial \tau} \right)$$	$$\frac{1}{\xi}F_{\tau r}= -\frac{\partial}{\partial r} \left( \frac{\partial \psi}{\partial \xi} \right)$$