User:DarkDeity5/sandbox

Fields & potentials
$$\left( \mathbf E, \mathbf B \right) = - \nabla \phi_\mathrm{e, m} - {\partial \over \partial \tau} \mathbf A_\mathrm{e, m} - \nabla \times \mathbf A_\mathrm{m, -e}$$

$$\left( \mathbf E, \mathbf B \right) = - \left( \nabla \phi_\mathrm{e, m} + {\partial \over \partial \tau} \mathbf A_\mathrm{e, m} + \nabla \times \mathbf A_\mathrm{m, -e} \right)$$

$$- \left( \mathbf E, \mathbf B \right) = \nabla \phi_\mathrm{e, m} + {\partial \over \partial \tau} \mathbf A_\mathrm{e, m} + \nabla \times \mathbf A_\mathrm{m, -e}$$

$$- \left( \mathbf E, \mathbf B \right) = \nabla \phi_\mathrm{e, m} + \left( {\partial \over \partial \tau} + \nabla \times \mathbf \Theta \left( - {\pi \over 2} \right) \right) \mathbf A_\mathrm{e, m} $$

Style 1
$$\begin{align} \nabla \cdot \left( \mathbf E, \mathbf B \right) & = \rho_\mathrm{e, m} = - \nabla^2 \phi_\mathrm{e, m} - {\partial \over \partial \tau} \left( \nabla \cdot \mathbf A_\mathrm{e, m} \right) \\ \nabla \times \left( \mathbf B, - \mathbf E \right) - {\partial \over \partial \tau} \left( \mathbf E, \mathbf B \right) & = {1 \over c} \mathbf j_\mathrm{e, m} = \Box \mathbf A_\mathrm{e, m} + \nabla \left( {\partial \over \partial \tau} \phi_\mathrm{e, m} + \nabla \cdot \mathbf A_\mathrm{e, m} \right) \\ \text{In Lorenz gague:} & \\ {\partial \over \partial \tau} \phi_\mathrm{e, m} + \nabla \cdot \mathbf A_\mathrm{e, m} & = 0 \\ \nabla \cdot \left( \mathbf E, \mathbf B \right) & = \rho_\mathrm{e, m} = \Box \phi_\mathrm{e, m} \\ \nabla \times \left( \mathbf B, - \mathbf E \right) - {\partial \over \partial \tau} \left( \mathbf E, \mathbf B \right) & = {1 \over c} \mathbf j_\mathrm{e, m} = \Box \mathbf A_\mathrm{e, m} \\ \end{align}$$

Style 2
$$\begin{array}{rclcl} \nabla \cdot \left( \mathbf E, \mathbf B \right) &=& \rho_\mathrm{e, m} &=& - \nabla^2 \phi_\mathrm{e, m} - \partial_\tau \left( \nabla \cdot \mathbf A_\mathrm{e, m} \right) \\ \nabla \times \left( \mathbf B, - \mathbf E \right) - \partial_\tau \left( \mathbf E, \mathbf B \right) &=& c^{-1} \mathbf j_\mathrm{e, m} &=& \Box \mathbf A_\mathrm{e, m} + \nabla \left( \partial_\tau \phi_\mathrm{e, m} + \nabla \cdot \mathbf A_\mathrm{e, m} \right) \\ \text{In Lorenz gague:} \\ \partial_\tau \phi_\mathrm{e, m} + \nabla \cdot \mathbf A_\mathrm{e, m} &=& 0 \\ \nabla \cdot \left( \mathbf E, \mathbf B \right) &=& \rho_\mathrm{e, m} &=& \Box \phi_\mathrm{e, m} \\ \nabla \times \left( \mathbf B, - \mathbf E \right) - \partial_\tau \left( \mathbf E, \mathbf B \right) &=& c^{-1} \mathbf j_\mathrm{e, m} &=& \Box \mathbf A_\mathrm{e, m} \\ \end{array}$$

Equations combined
$$\begin{align} \left( \mathbf E, \mathbf B \right) & = - \nabla \phi_\mathrm{e, m} - {\partial \over \partial \tau} \mathbf A_\mathrm{e, m} - \nabla \times \mathbf A_\mathrm{m, -e} \\ 0 & = {\partial \over \partial \tau} \phi_\mathrm{e, m} + \nabla \cdot \mathbf A_\mathrm{e, m} \\ \nabla \cdot \left( \mathbf E, \mathbf B \right) & = \rho_\mathrm{e, m} = \Box \phi_\mathrm{e, m} \\ \nabla \times \left( \mathbf B, - \mathbf E \right) - {\partial \over \partial \tau} \left( \mathbf E, \mathbf B \right) & = {1 \over c} \mathbf j_\mathrm{e, m} = \Box \mathbf A_\mathrm{e, m} \\ \end{align}$$