Problem 20
Write the equation $$ F_{\alpha \beta}=A_{\beta, \alpha}-A_{\alpha, \beta}=\frac{\partial A_{\beta}}{\partial x^{\alpha}}-\frac{\partial A_{\alpha}}{\partial x^{\beta}} $$ in terms of \(\mathbf{E}, \mathbf{B}\), and vector and scalar potentials.
Problem 21
With \(\mathbf{F}=\frac{1}{2} F_{\alpha \beta} d x^{\alpha} \wedge d x^{\beta}\) and \(\mathbf{J}=J_{\gamma} d x^{\gamma}\), show that \(d * \mathbf{F}=4 \pi(* \mathbf{J})\) takes the following form in components: $$ \frac{\partial F^{\alpha \beta}}{\partial x^{\beta}}=4 \pi J^{\alpha}, $$ where indices are raised and lowered by diag \((-1,-1,-1,1)\).
Problem 24
Show that current conservation is an automatic consequence of Maxwell's inhomogeneous equation \(d * \mathbf{F}=4 \pi(* \mathbf{J})\).
