The Eigenvector Fields of $\mathcal A$

The task of identifying and determining those eigenvector fields of $\mathcal A$ which have nonzero eigenvalues is facilitated by the fact that Eq.(6.64) demands that they be divergenceless. The three vector fields in Eq.() do satisfy this condition. That they span the eigenspaces in the range of $\mathcal A$ is verified by three explicit calculations, one for each the three eigenvector fields. Inserting them into the Maxwell wave equation, one finds that their linear independence results in the following three independent vector equations:

$${\mathcal A} \left[\!\! \begin{array}{r} 0\\ 0\\ \partial_y \\ -\partial_x \end{array}\!\! \right]\Phi^{TE} = \left[\!\! \begin{array}{r} 0\\ 0\\ \partial_y\\ -\partial_x \e... ...\! \begin{array}{c} 0\\ 0\\ \partial_y\\ -\partial_x \end{array} \right] S^{TE}$$ (668)

$${\mathcal A} \left[\!\! \begin{array}{r} -\partial_z\\ \partial_t\\ 0\\ 0 \end{array}\!\! \right]\Phi^{TM} = \left[\!\! \begin{array}{r} -\partial_z\\ \partial_t\\ 0\\ 0 \e... ... \begin{array}{r} -\partial_z \\ \partial_t \\ 0\\ 0 \end{array} \right] S^{TM}$$ (669)
and

$${\mathcal A} \left[\!\! \begin{array}{r} -\partial_t \Phi\\ \partial_z \Phi\\ \partial_x \Psi\\ \partial_y \Psi \end{array}\!\! \right] = \left[\!\!\! \begin{array}{c} \left(\!\! \begin{array}{r} -\par... ...rray}{r} \partial_x\\ \partial_y \end{array} \right)I \end{array} \!\!\!\right]$$ (670)

What is remarkable about these equations is that each of them is an integrable system which can be integrated by inspection. Doing so results in what in linear algebra corresponds to the three equations (6.58)-() on page  for the eigenvector amplitudes $c_i$ . Here. however, the result is three scalar wave equations for the scalar fields $\Phi^{TE},\Phi^{TM},\Psi$ and $\Phi$ , namely⁶⁸

$$\left( \partial^2_x+\partial^2_y+\partial^2_z-\partial^2_t \right) \Phi^{TE} =-4\pi S^{TE}$$ (671)

$$\left( \partial^2_x+\partial^2_y+\partial^2_z-\partial^2_t \right) \Phi^{TM} =-4\pi S^{TM}$$ (672)
and

$$(\partial_x^2+\partial_y^2)(\Phi-\Psi) =-4\pi J$$ (673)

$$(\partial_z^2-\partial_t^2)(\Phi-\Psi) =+4\pi I ~.$$ (674)

The last two equations are the TEM equations, a Poisson and a wave equation, for one and the same quantity, the difference $\Phi -\Psi$ . These two equations are consistent, and hence integrable, because the TEM source scalars $I$ and $J$ are guaranteed to satisfy Eq.(6.67) on page . Put differently, any two of the three equations (), (6.74), and (6.67) imply the third.

Footnotes

... namely⁶⁸

The superscripts TE, TM, as well as TEM below are acronyms which stand for transverse electric, transverse magnetic, and transeverse electric magnetic The justification for these apellations are given on Pages , , and , repectively.