Then by~\Eq{(73.1)} \begin{alignat*}{2} F_{14} &= \frac{\dd\kappa_{1}}{\dd x_{4}} - \frac{\dd\kappa_{4}}{\dd x_{1}} &&= \frac{\dd(-F)}{\dd t} - \frac{\dd\Phi}{\dd x} = X, \\ F_{23} &= \frac{\dd\kappa_{2}}{\dd x_{3}} - \frac{\dd\kappa_{3}}{\dd x_{2}} &&= \frac{\dd(-G)}{\dd z} - \frac{\dd(-H)}{\dd y} = \alpha. \end{alignat*} \PageSep{172} Accordingly the electric and magnetic forces together form the curl of the electromagnetic potential. The complete scheme for~$F_{\mu\nu}$ is \[ \begin{array}[t]{r} F_{\mu\nu} \\ \MuNuarrow \end{array} = \begin{array}[t]{@{}r@{\Add{,}\quad}r@{\Add{,}\quad}r@{\Add{,}\quad}r@{}} 0 & -\gamma & \beta & -X\Add{,} \\ \gamma & 0 & -\alpha & -Y\Add{,} \\ -\beta & \alpha & 0 & -Z\Add{,} \\ X & Y & Z & 0. \\ \end{array} \Tag{(73.41)} \] Using Galilean values of~$g^{\mu\nu}$ to raise the two suffixes, \[ F^{\mu\nu} = \begin{array}[t]{@{}r@{\Add{,}\quad}r@{\Add{,}\quad}r@{\Add{,}\quad}r@{}} 0 & -\gamma & \beta & \Neg X\Add{,} \\ \gamma & 0 & -\alpha & Y\Add{,} \\ -\beta & \alpha & 0 & Z\Add{,} \\ -X & -Y & -Z & 0. \\ \end{array} \Tag{(73.42)} \] Let $\rho$~be the density of electric charge and $\sigma_{x}$, $\sigma_{y}$, $\sigma_{z}$ the density of electric current. We set \index{Charge-and-current vector}% \index{Current, electric}% \index{J@$J^{\mu}$ (charge-and-current vector)}% \[ J^{\mu} = (\sigma_{x}, \sigma_{y}, \sigma_{z}, \rho). \Tag{(73.5)} \] Here again we must not assume that the components of~$J^{\mu}$ will be recognised experimentally as electric charge and current-density except in the original system of coordinates. The universally accepted laws of the electromagnetic field are those given by Maxwell. Maxwell's equations are \index{Maxwell's equations}% \begin{gather*} \frac{\dd Z}{\dd y} - \frac{\dd Y}{\dd z} = -\frac{\dd\alpha}{\dd t},\ \frac{\dd X}{\dd z} - \frac{\dd Z}{\dd x} = -\frac{\dd\beta}{\dd t},\ \frac{\dd Y}{\dd x} - \frac{\dd X}{\dd y} = -\frac{\dd\gamma}{\dd t}, \Tag{(73.61)}\displaybreak[0] \\ % \frac{\dd\gamma}{\dd y} - \frac{\dd\beta}{\dd z} = \frac{\dd X}{\dd t} + \sigma_{x},\ \frac{\dd\alpha}{\dd z} - \frac{\dd\gamma}{\dd x} = \frac{\dd Y}{\dd t} + \sigma_{y},\ \frac{\dd\beta}{\dd x} - \frac{\dd\alpha}{\dd y} = \frac{\dd Z}{\dd t} + \sigma_{z}, \Tag{(73.62)}\displaybreak[0] \\ % \frac{\dd X}{\dd x} + \frac{\dd Y}{\dd y} + \frac{\dd Z}{\dd z} = \rho, \Tag{(73.63)}\displaybreak[0] \\ \frac{\dd \alpha}{\dd x} + \frac{\dd \beta}{\dd y} + \frac{\dd \gamma}{\dd z} = 0. \Tag{(73.64)} \end{gather*} The Heaviside-Lorentz unit of charge is used so that the factor~$4\pi$ does not appear. The velocity of light is as usual taken to be unity. Specific inductive capacity and magnetic permeability are merely devices employed in obtaining macroscopic equations, and do not occur in the exact theory.